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Domain > limit-calculator.org
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Date
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2018-09-20
46.4.24.196
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ClassC
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2026-02-16
212.102.46.118
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ClassC
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Port 80
HTTP/1.1 301 Moved PermanentlyDate: Mon, 16 Feb 2026 14:53:11 GMTContent-Type: text/htmlContent-Length: 166Connection: keep-aliveServer: BunnyCDN-WA1-1120CDN-PullZone: 4725598CDN-RequestCountryCode: USLocation: https://limit-calculator.org/CDN-RequestId: a1dc3aac102228d929f1df548baed0b0CDN-RequestTime: 0 html>head>title>301 Moved Permanently/title>/head>body>center>h1>301 Moved Permanently/h1>/center>hr>center>openresty/center>/body>/html>
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HTTP/1.1 200 OKDate: Mon, 16 Feb 2026 14:53:11 GMTContent-Type: text/html; charsetutf-8Transfer-Encoding: chunkedConnection: keep-aliveVary: Accept-EncodingServer: BunnyCDN-WA1-1120CDN-PullZone: 4725598CDN-RequestCountryCode: USCache-Control: public, max-age259200Content-Security-Policy: frame-ancestors self https://limit-calculator.org;CDN-ProxyVer: 1.43CDN-RequestPullCode: 200CDN-RequestPullSuccess: TrueCDN-EdgeStorageId: 1120CDN-CachedAt: 02/14/2026 00:24:12CDN-Status: 200CDN-RequestTime: 0CDN-RequestId: 508e187a7c41a8f5358585fad4ff9096CDN-Cache: HIT !DOCTYPE html>html langen>head> meta charsetUTF-8> meta nameviewport contentwidthdevice-width, initial-scale1.0> title>Limit Calculator/title> meta namedescription contentA free calculus calculator to evaluate the limit of a function. Solves limits as x approaches a specific number or infinity, with step-by-step explanations.> !-- --> !-- START: SEO and Language Tags --> !-- --> link relcanonical hrefhttps://limit-calculator.org/ /> link relalternate hreflangx-default hrefhttps://limit-calculator.org/ /> link relalternate hreflangen hrefhttps://limit-calculator.org/ /> !-- --> !-- END: SEO and Language Tags --> !-- --> script typeapplication/ld+json> {@context: https://schema.org, @type: WebApplication, applicationCategory: EducationalApplication, description: A calculus calculator for evaluating limits of functions, including one-sided limits and limits at infinity., name: Limit Calculator} /script> script typeapplication/ld+json> { @context: https://schema.org, @type: FAQPage, mainEntity: { @type: Question, name: What is a limit in calculus?, acceptedAnswer: { @type: Answer, text: A limit is the value that a function "approaches" as the input approaches some value. It is a fundamental concept used to define continuity, derivatives, and integrals. } }, { @type: Question, name: What is an indeterminate form?, acceptedAnswer: { @type: Answer, text: An indeterminate form is an expression like 0/0 or ∞/∞, where simply plugging in the value does not determine the limit. Techniques like L'Hopital's Rule or algebraic manipulation are needed to resolve them. } }, { @type: Question, name: What does it mean if a limit does not exist?, acceptedAnswer: { @type: Answer, text: A limit does not exist if the function approaches different values from the left and the right, if the function grows without bound (approaches ±∞), or if it oscillates infinitely. } } } /script> !-- Dynamic Favicon Logic --> style> /* --- General & Responsive Setup --- */ /* 1. THE CRITICAL FIX: Add this block */ *, *::before, *::after { box-sizing: border-box; } :root { --primary-color: #007bff; --primary-hover: #0056b3; --text-color: #333; --border-color: #e0e0e0; --background-color: #fff; --footer-bg: #222; --footer-text: #ddd; } /* 2. 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Solves limits as x approaches a specific number or infinity, with step-by-step explanations.> meta nametwitter:image contenthttps://limit-calculator.org/static/ogimages/limit-calculator.org.png> meta nameadmaven-placement contentBqjwFqTYE> script typetext/javascript> (function(c,l,a,r,i,t,y){ caca||function(){(ca.qca.q||).push(arguments)}; tl.createElement(r);t.async1;t.srchttps://www.clarity.ms/tag/+i; yl.getElementsByTagName(r)0;y.parentNode.insertBefore(t,y); })(window, document, clarity, script, uebksgowh4); /script>!-- Matomo -->script> var _paq window._paq window._paq || ; /* tracker methods like setCustomDimension should be called before trackPageView */ _paq.push(trackPageView); _paq.push(enableLinkTracking); (function() { var u//tracking.limit-calculator.org/; _paq.push(setTrackerUrl, u+matomo.php); _paq.push(setSiteId, 879); var ddocument, gd.createElement(script), sd.getElementsByTagName(script)0; g.asynctrue; g.srcu+matomo.js; s.parentNode.insertBefore(g,s); })();/script>!-- End Matomo Code -->/head>body>header classmain-header> div classcontainer> div classheader-brand-container> a href/ classheader-brand> span classheader-brand-emoji>→/span> span>Limit Calculator/span> /a> /div> div classheader-right-side> /div> /div> !-- Ad Slots inside Header Container --> /header> main classcontainer> div classcontent-wrapper> !-- --> !-- LOGIC SWITCH: INNER PAGE VS LANDING PAGE --> !-- --> !-- *** LANDING PAGE LAYOUT *** --> section classcontent-section intro-section> h1>Online Limit Calculator with Steps/h1> p>The Limit Calculator is a powerful tool designed to help you evaluate the limit of a function at a given point. 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border-radius: 8px; margin-top: 25px;}.limit-calculator .visualization-container { display: flex; justify-content: center; align-items: center; padding: 20px; background-color: #fff; border-radius: 8px;}.limit-calculator .visualization-container canvas { max-width: 100%; height: auto;}/* Calculation Steps */.limit-calculator .calculation-steps { background-color: #f8f9fa; padding: 20px; border-radius: 8px; margin-top: 25px; border-left: 4px solid #2ecc71;}.limit-calculator .steps-content { line-height: 1.6;}.limit-calculator .steps-content p { margin: 8px 0;}/* Formula Section (Limit Properties) */.limit-calculator .formula-section { background-color: #f0f7fb; padding: 20px; border-radius: 8px; margin-top: 25px;}.limit-calculator .convergence-content { line-height: 1.6;}.limit-calculator .convergence-content p { margin: 8px 0;}/* Calculator Footer & Theory Section */.limit-calculator .calculator-footer { margin-top: 40px; padding-top: 20px; border-top: 1px solid #e8e8e8;}.limit-calculator .theory-section { background-color: #f8f9fa; padding: 20px; border-radius: 8px; margin-bottom: 20px;}.limit-calculator .theory-section h4 { color: #2980b9; margin-top: 0; margin-bottom: 15px;}.limit-calculator .theory-section p { line-height: 1.6; margin-bottom: 15px;}.limit-calculator .theory-subsection { margin-top: 20px; margin-bottom: 20px;}.limit-calculator .theory-subsection h5 { color: #2980b9; margin-top: 0; margin-bottom: 10px; font-size: 16px;}.limit-calculator .theory-subsection ul,.limit-calculator .theory-subsection ol { margin: 0; padding-left: 20px;}.limit-calculator .theory-subsection li { margin-bottom: 8px; line-height: 1.5;}.limit-calculator .theory-subsection p { margin: 10px 0;}/* Remove number input spinners for Firefox & Webkit */.limit-calculator inputtypenumber { -moz-appearance: textfield;}.limit-calculator inputtypenumber::-webkit-outer-spin-button,.limit-calculator inputtypenumber::-webkit-inner-spin-button { -webkit-appearance: none; margin: 0;}/* Radio & Checkbox Custom Styling */.limit-calculator inputtyperadio,.limit-calculator inputtypecheckbox { accent-color: #2980b9;}/* Responsive Styling */@media (max-width: 767px) { .limit-calculator .input-row { flex-direction: column; gap: 10px; } .limit-calculator .input-group { min-width: 100%; } .limit-calculator .btn-group { flex-direction: column; } .limit-calculator .results-grid { grid-template-columns: 1fr; } .limit-calculator .result-card.primary { grid-column: 1; } .limit-calculator .visualization-container canvas { width: 100% !important; height: auto !important; } .limit-calculator .result-value { font-size: 24px; }}@media (min-width: 768px) and (max-width: 991px) { .limit-calculator .results-grid { grid-template-columns: repeat(2, 1fr); } .limit-calculator .result-card.primary { grid-column: 1 / -1; }}@media (min-width: 992px) { .limit-calculator .theory-section { padding: 30px; } .limit-calculator .results-grid { grid-template-columns: repeat(4, 1fr); } .limit-calculator .result-card.primary { grid-column: 1 / -1; }}/style>style> /* Styling for parent elements containing formulas */ .formula-container { display: block; overflow-x: auto; margin-bottom: 1rem; padding: 0.5rem; background-color: #f9f9f9; border: 1px solid #ddd; border-radius: 4px; }/style> script>// // Limit Calculation Functions// function calculateLimit() { const functionType document.getElementById(function-type).value; const limitPointStr document.getElementById(limit-point).value.trim(); const direction document.getElementById(limit-direction).value; const decimalPlaces parseInt(document.getElementById(decimal-places).value); const useNumerical document.getElementById(numerical-method).checked; try { // Parse limit point const limitPoint parseLimitPoint(limitPointStr); // Get function expression const functionExpr getFunctionExpression(functionType); if (!functionExpr) { throw new Error(Please enter a function expression); } // Calculate limits let limitValue, leftLimit, rightLimit, functionValueAtPoint; let steps ; let properties ; if (direction both) { leftLimit evaluateLimit(functionExpr, limitPoint, left, useNumerical); rightLimit evaluateLimit(functionExpr, limitPoint, right, useNumerical); // Two-sided limit exists only if both one-sided limits exist and are equal if (leftLimit.value ! null && rightLimit.value ! null && Math.abs(leftLimit.value - rightLimit.value) 1e-6) { limitValue leftLimit.value; } else if (leftLimit.value ! null && rightLimit.value ! null) { limitValue DNE (Does Not Exist); } else { limitValue leftLimit.value ! null ? leftLimit.value : rightLimit.value; } } else if (direction left) { leftLimit evaluateLimit(functionExpr, limitPoint, left, useNumerical); limitValue leftLimit.value; } else if (direction right) { rightLimit evaluateLimit(functionExpr, limitPoint, right, useNumerical); limitValue rightLimit.value; } // Try to evaluate function at the point (if not infinity) if (isFinite(limitPoint)) { try { functionValueAtPoint evaluateFunctionAt(functionExpr, limitPoint); } catch (e) { functionValueAtPoint Undefined; } } else { functionValueAtPoint N/A; } // Check continuity let continuity N/A; if (isFinite(limitPoint) && direction both) { if (leftLimit.value ! null && rightLimit.value ! null && Math.abs(leftLimit.value - rightLimit.value) 1e-6) { if (functionValueAtPoint ! Undefined && Math.abs(functionValueAtPoint - limitValue) 1e-6) { continuity Continuous; } else if (functionValueAtPoint Undefined) { continuity Removable Discontinuity; } else { continuity Removable Discontinuity; } } else { continuity Jump Discontinuity; } } // Generate solution steps steps generateSolutionSteps(functionExpr, limitPoint, direction, limitValue, leftLimit, rightLimit, functionType); // Generate limit properties properties generateLimitProperties(functionExpr, limitPoint, functionType); // Generate approaching values const approachingValues generateApproachingValues(functionExpr, limitPoint, direction); return { functionExpr: functionExpr, functionType: functionType, limitPoint: limitPoint, limitPointStr: limitPointStr, direction: direction, limitValue: limitValue, leftLimit: leftLimit ? leftLimit.value : null, rightLimit: rightLimit ? rightLimit.value : null, functionValueAtPoint: functionValueAtPoint, continuity: continuity, steps: steps, properties: properties, approachingValues: approachingValues, decimalPlaces: decimalPlaces }; } catch (error) { throw error; }}function parseLimitPoint(str) { str str.toLowerCase().trim(); if (str inf || str ∞ || str infinity) { return Infinity; } else if (str -inf || str -∞ || str -infinity) { return -Infinity; } else { const num parseFloat(str); if (isNaN(num)) { throw new Error(Invalid limit point. Use a number, inf, or -inf); } return num; }}function getFunctionExpression(functionType) { switch (functionType) { case polynomial: return document.getElementById(polynomial-expression).value.trim(); case rational: const num document.getElementById(numerator-expression).value.trim(); const den document.getElementById(denominator-expression).value.trim(); if (!num || !den) return ; return `(${num}) / (${den})`; case trigonometric: return document.getElementById(trig-expression).value.trim(); case exponential: return document.getElementById(exponential-expression).value.trim(); case logarithmic: return document.getElementById(logarithmic-expression).value.trim(); case radical: return document.getElementById(radical-expression).value.trim(); case piecewise: return piecewise; // Special handling case custom: return document.getElementById(custom-expression).value.trim(); default: return ; }}function evaluateLimit(expression, point, direction, useNumerical) { if (expression piecewise) { return evaluatePiecewiseLimit(point, direction); } // Use numerical approximation if (useNumerical) { return numericalLimit(expression, point, direction); } // Try direct substitution first try { if (isFinite(point)) { const directValue evaluateFunctionAt(expression, point); if (isFinite(directValue)) { return { value: directValue, method: direct }; } } } catch (e) { // Continue to numerical method } // Fall back to numerical return numericalLimit(expression, point, direction);}function evaluatePiecewiseLimit(point, direction) { const leftExpr document.getElementById(piecewise-left).value.trim(); const rightExpr document.getElementById(piecewise-right).value.trim(); const atExpr document.getElementById(piecewise-at).value.trim(); if (direction left) { return numericalLimit(leftExpr, point, left); } else if (direction right) { return numericalLimit(rightExpr, point, right); } return { value: null, method: piecewise };}function numericalLimit(expression, point, direction) { try { const node math.parse(expression); const compiled node.compile(); const epsilon 1e-6; const steps 10; let values ; if (!isFinite(point)) { // Limit at infinity const sign point > 0 ? 1 : -1; for (let i 1; i steps; i++) { const x sign * Math.pow(10, i); try { const y compiled.evaluate({ x: x }); if (isFinite(y)) { values.push(y); } } catch (e) { // Skip invalid values } } } else { // Limit at finite point for (let i 1; i steps; i++) { const delta epsilon * Math.pow(10, -i/2); let x; if (direction left) { x point - delta; } else { x point + delta; } try { const y compiled.evaluate({ x: x }); if (isFinite(y)) { values.push(y); } } catch (e) { // Skip invalid values } } } if (values.length 0) { return { value: null, method: numerical }; } // Check if values are converging const lastValue valuesvalues.length - 1; // Check for convergence to infinity if (Math.abs(lastValue) > 1e10) { return { value: lastValue > 0 ? Infinity : -Infinity, method: numerical }; } return { value: lastValue, method: numerical }; } catch (error) { return { value: null, method: numerical, error: error.message }; }}function evaluateFunctionAt(expression, x) { const node math.parse(expression); const compiled node.compile(); return compiled.evaluate({ x: x });}function generateApproachingValues(expression, point, direction) { const values ; if (expression piecewise) { // Handle piecewise separately return generatePiecewiseValues(point, direction); } try { const node math.parse(expression); const compiled node.compile(); if (!isFinite(point)) { // Approaching infinity const sign point > 0 ? 1 : -1; const xValues 10, 100, 1000, 10000, 100000.map(v > sign * v); xValues.forEach(x > { try { const y compiled.evaluate({ x: x }); values.push({ x: x, fx: y, distance: Math.abs(x) }); } catch (e) { values.push({ x: x, fx: Undefined, distance: Math.abs(x) }); } }); } else { // Approaching finite point let xValues ; if (direction left) { xValues 0.1, 0.01, 0.001, 0.0001, 0.00001.map(d > point - d); } else if (direction right) { xValues 0.1, 0.01, 0.001, 0.0001, 0.00001.map(d > point + d); } else { // Both directions xValues point - 0.1, point - 0.01, point - 0.001, point + 0.001, point + 0.01, point + 0.1 ; } xValues.forEach(x > { try { const y compiled.evaluate({ x: x }); values.push({ x: x, fx: y, distance: Math.abs(x - point) }); } catch (e) { values.push({ x: x, fx: Undefined, distance: Math.abs(x - point) }); } }); } } catch (e) { // Return empty if expression is invalid } return values;}function generatePiecewiseValues(point, direction) { const leftExpr document.getElementById(piecewise-left).value.trim(); const rightExpr document.getElementById(piecewise-right).value.trim(); const values ; try { const leftNode math.parse(leftExpr); const leftCompiled leftNode.compile(); const rightNode math.parse(rightExpr); const rightCompiled rightNode.compile(); const leftValues point - 0.1, point - 0.01, point - 0.001; const rightValues point + 0.001, point + 0.01, point + 0.1; leftValues.forEach(x > { try { const y leftCompiled.evaluate({ x: x }); values.push({ x: x, fx: y, distance: Math.abs(x - point) }); } catch (e) { values.push({ x: x, fx: Undefined, distance: Math.abs(x - point) }); } }); rightValues.forEach(x > { try { const y rightCompiled.evaluate({ x: x }); values.push({ x: x, fx: y, distance: Math.abs(x - point) }); } catch (e) { values.push({ x: x, fx: Undefined, distance: Math.abs(x - point) }); } }); } catch (e) { // Return empty } return values;}// // Solution Steps Generation// function generateSolutionSteps(expr, point, direction, limitValue, leftLimit, rightLimit, functionType) { const steps ; steps.push(`p>strong>Given:/strong>/p>`); steps.push(`p>Function: f(x) ${expr}/p>`); const pointStr isFinite(point) ? point : (point > 0 ? ∞ : -∞); if (direction both) { steps.push(`p>Calculate: lim(x→${pointStr}) f(x)/p>`); } else if (direction left) { steps.push(`p>Calculate: lim(x→${pointStr}⁻) f(x)/p>`); } else { steps.push(`p>Calculate: lim(x→${pointStr}⁺) f(x)/p>`); } steps.push(`p>strong>Step 1: Identify the limit type/strong>/p>`); if (!isFinite(point)) { steps.push(`p>This is a limit at infinity (x → ${pointStr})/p>`); } else { steps.push(`p>This is a finite limit (x → ${point})/p>`); // Try direct substitution steps.push(`p>strong>Step 2: Try direct substitution/strong>/p>`); try { const directValue evaluateFunctionAt(expr, point); if (isFinite(directValue)) { steps.push(`p>f(${point}) ${directValue.toFixed(6)}/p>`); steps.push(`p>The function is defined at x ${point}, so we can use direct substitution./p>`); } } catch (e) { steps.push(`p>Direct substitution gives an indeterminate form or undefined value./p>`); steps.push(`p>We need to use other techniques (factoring, rationalization, or LHôpitals rule)./p>`); } } if (direction both) { steps.push(`p>strong>Step 3: Evaluate one-sided limits/strong>/p>`); if (leftLimit && leftLimit.value ! null) { const leftStr formatLimitValue(leftLimit.value); steps.push(`p>Left-hand limit: lim(x→${pointStr}⁻) f(x) ${leftStr}/p>`); } if (rightLimit && rightLimit.value ! null) { const rightStr formatLimitValue(rightLimit.value); steps.push(`p>Right-hand limit: lim(x→${pointStr}⁺) f(x) ${rightStr}/p>`); } if (leftLimit && rightLimit && leftLimit.value ! null && rightLimit.value ! null) { if (Math.abs(leftLimit.value - rightLimit.value) 1e-6) { steps.push(`p>Since both one-sided limits exist and are equal, the two-sided limit exists./p>`); } else { steps.push(`p>The one-sided limits are different, so the two-sided limit does not exist./p>`); } } } steps.push(`p>strong>Final Result:/strong>/p>`); const resultStr formatLimitValue(limitValue); steps.push(`p>lim(x→${pointStr}) f(x) ${resultStr}/p>`); return steps;}function generateLimitProperties(expr, point, functionType) { const properties ; properties.push(`p>strong>Limit Properties Applied:/strong>/p>`); if (functionType polynomial) { properties.push(`p>• strong>Polynomial Limit:/strong> For polynomial functions, the limit can be found by direct substitution./p>`); } else if (functionType rational) { properties.push(`p>• strong>Quotient Rule:/strong> limf(x)/g(x) limf(x) / limg(x) if limg(x) ≠ 0/p>`); properties.push(`p>• If direct substitution gives 0/0, factor and simplify, or use LHôpitals rule./p>`); } else if (functionType trigonometric) { properties.push(`p>• strong>Trigonometric Limits:/strong> Special limits like lim(x→0) sin(x)/x 1/p>`); } else if (functionType exponential) { properties.push(`p>• strong>Exponential Limits:/strong> lim(x→0) (e^x - 1)/x 1/p>`); } else if (functionType logarithmic) { properties.push(`p>• strong>Logarithmic Limits:/strong> lim(x→0) ln(1+x)/x 1/p>`); } if (!isFinite(point)) { properties.push(`p>• strong>Limits at Infinity:/strong> Examine the highest degree terms in numerator and denominator./p>`); } properties.push(`p>• strong>Numerical Approximation:/strong> Values calculated by approaching the limit point from the specified direction./p>`); return properties;}function formatLimitValue(value) { if (value null || value undefined) { return DNE (Does Not Exist); } if (value DNE (Does Not Exist)) { return value; } if (value Infinity) { return ∞; } if (value -Infinity) { return -∞; } if (typeof value number) { return value.toFixed(6); } return String(value);}// // Display Functions// function formatNumber(value, precision) { if (value null || value undefined || value Undefined) { return Undefined; } if (value Infinity) { return ∞; } if (value -Infinity) { return -∞; } if (typeof value string) { return value; } if (!isFinite(value)) { return Undefined; } return Number(value).toFixed(precision);}function displayResults(result) { const precision result.decimalPlaces; // Display main limit value const limitStr formatLimitValue(result.limitValue); document.getElementById(limit-value).textContent limitStr; const pointStr isFinite(result.limitPoint) ? result.limitPoint : (result.limitPoint > 0 ? ∞ : -∞); let directionSymbol ; if (result.direction left) directionSymbol ⁻; if (result.direction right) directionSymbol ⁺; document.getElementById(limit-notation).textContent `lim(x→${pointStr}${directionSymbol}) f(x)`; // Display one-sided limits if both direction if (result.direction both && isFinite(result.limitPoint)) { document.getElementById(one-sided-limits).classList.remove(hidden); document.getElementById(left-limit).textContent formatNumber(result.leftLimit, precision); document.getElementById(right-limit).textContent formatNumber(result.rightLimit, precision); document.getElementById(function-value).textContent formatNumber(result.functionValueAtPoint, precision); document.getElementById(continuity-status).textContent result.continuity; } else { document.getElementById(one-sided-limits).classList.add(hidden); } // Display limit notation with MathJax displayLimitNotation(result); // Display approaching values table if (document.getElementById(show-table).checked) { document.getElementById(values-table-container).classList.remove(hidden); displayValuesTable(result); } else { document.getElementById(values-table-container).classList.add(hidden); } // Display graph if (document.getElementById(show-graph).checked) { document.getElementById(graph-container).classList.remove(hidden); drawLimitGraph(result); } else { document.getElementById(graph-container).classList.add(hidden); } // Display solution steps if (document.getElementById(show-steps).checked) { document.getElementById(solution-steps).classList.remove(hidden); document.getElementById(steps-content).innerHTML result.steps.join(); } else { document.getElementById(solution-steps).classList.add(hidden); } // Display limit properties document.getElementById(properties-content).innerHTML result.properties.join(); // Show results container document.getElementById(result-container).classList.remove(hidden); // Scroll to results document.getElementById(result-container).scrollIntoView({ behavior: smooth, block: start });}function displayLimitNotation(result) { const pointStr isFinite(result.limitPoint) ? result.limitPoint : (result.limitPoint > 0 ? \\infty : -\\infty); let directionStr ; if (result.direction left) directionStr ^-; if (result.direction right) directionStr ^+; const limitValueStr formatLimitValue(result.limitValue); const limitValueLatex limitValueStr.replace(∞, \\infty).replace(-∞, -\\infty); const latex `$$\\lim_{x \\to ${pointStr}${directionStr}} f(x) ${limitValueLatex}$$`; const notationDiv document.getElementById(notation-formula); notationDiv.innerHTML latex; if (window.MathJax) { MathJax.typesetPromise(notationDiv).catch((err) > console.log(err)); }}function displayValuesTable(result) { const tableBody document.getElementById(values-table-body); tableBody.innerHTML ; result.approachingValues.forEach(item > { const row document.createElement(tr); row.innerHTML ` td>${formatNumber(item.x, result.decimalPlaces)}/td> td>${formatNumber(item.fx, result.decimalPlaces)}/td> td>${formatNumber(item.distance, result.decimalPlaces)}/td> `; tableBody.appendChild(row); });}// // Graph Functions// let limitChart null;function drawLimitGraph(result) { const canvas document.getElementById(limit-graph-canvas); const ctx canvas.getContext(2d); // Destroy existing chart if (limitChart) { limitChart.destroy(); } const expr result.functionExpr; const point result.limitPoint; // Determine x range let xMin, xMax; if (!isFinite(point)) { xMin point > 0 ? 1 : -100; xMax point > 0 ? 100 : -1; } else { const range 5; xMin point - range; xMax point + range; } // Generate function points const numPoints 200; const step (xMax - xMin) / numPoints; const functionPoints ; for (let i 0; i numPoints; i++) { const x xMin + i * step; // Skip the exact limit point for certain function types if (isFinite(point) && Math.abs(x - point) 0.01) { continue; } try { let y; if (expr piecewise) { y evaluatePiecewiseAt(x, point); } else { y evaluateFunctionAt(expr, x); } if (isFinite(y) && Math.abs(y) 1000) { functionPoints.push({ x: x, y: y }); } } catch (e) { // Skip invalid points } } // Prepare datasets const datasets { label: f(x), data: functionPoints, borderColor: rgb(41, 128, 185), backgroundColor: rgba(41, 128, 185, 0.1), borderWidth: 2, pointRadius: 0, showLine: true, tension: 0.1 } ; // Add limit point if finite if (isFinite(point) && result.limitValue ! DNE (Does Not Exist) && isFinite(result.limitValue)) { datasets.push({ label: Limit Point, data: { x: point, y: result.limitValue }, borderColor: rgb(231, 76, 60), backgroundColor: rgb(231, 76, 60), pointRadius: 8, pointStyle: circle, showLine: false }); } limitChart new Chart(ctx, { type: scatter, data: { datasets: datasets }, options: { responsive: true, maintainAspectRatio: true, plugins: { title: { display: true, text: `Graph of f(x) near x ${isFinite(point) ? point : (point > 0 ? ∞ : -∞)}`, font: { size: 16 } }, legend: { display: true, position: top } }, scales: { x: { type: linear, title: { display: true, text: x } }, y: { type: linear, title: { display: true, text: f(x) } } } } });}function evaluatePiecewiseAt(x, point) { const leftExpr document.getElementById(piecewise-left).value.trim(); const rightExpr document.getElementById(piecewise-right).value.trim(); const atExpr document.getElementById(piecewise-at).value.trim(); if (x point) { return evaluateFunctionAt(leftExpr, x); } else if (x > point) { return evaluateFunctionAt(rightExpr, x); } else { if (atExpr) { return evaluateFunctionAt(atExpr, x); } throw new Error(Undefined at point); }}// // Validation Functions// function validateInputs() { const functionType document.getElementById(function-type).value; const limitPoint document.getElementById(limit-point).value.trim(); if (!limitPoint) { alert(Please enter a limit point.); return false; } // Check if function expression is entered const expr getFunctionExpression(functionType); if (!expr) { alert(Please enter a function expression.); return false; } // Validate piecewise function if (functionType piecewise) { const left document.getElementById(piecewise-left).value.trim(); const right document.getElementById(piecewise-right).value.trim(); if (!left || !right) { alert(Please enter both expressions for the piecewise function.); return false; } } return true;}// // UI Functions// function resetCalculator() { document.getElementById(function-type).value polynomial; document.getElementById(polynomial-expression).value ; document.getElementById(numerator-expression).value ; document.getElementById(denominator-expression).value ; document.getElementById(trig-expression).value ; document.getElementById(exponential-expression).value ; document.getElementById(logarithmic-expression).value ; document.getElementById(radical-expression).value ; document.getElementById(piecewise-left).value ; document.getElementById(piecewise-right).value ; document.getElementById(piecewise-at).value ; document.getElementById(custom-expression).value ; document.getElementById(limit-point).value 0; document.getElementById(limit-direction).value both; document.getElementById(decimal-places).value 6; document.getElementById(show-table).checked true; document.getElementById(show-graph).checked true; document.getElementById(show-steps).checked true; document.getElementById(numerical-method).checked true; updateFunctionTypeDisplay(); document.getElementById(result-container).classList.add(hidden); if (limitChart) { limitChart.destroy(); limitChart null; } window.scrollTo({ top: 0, behavior: smooth });}function updateFunctionTypeDisplay() { const functionType document.getElementById(function-type).value; document.getElementById(polynomial-section).classList.toggle(hidden, functionType ! polynomial); document.getElementById(rational-section).classList.toggle(hidden, functionType ! rational); document.getElementById(trigonometric-section).classList.toggle(hidden, functionType ! trigonometric); document.getElementById(exponential-section).classList.toggle(hidden, functionType ! exponential); document.getElementById(logarithmic-section).classList.toggle(hidden, functionType ! logarithmic); document.getElementById(radical-section).classList.toggle(hidden, functionType ! radical); document.getElementById(piecewise-section).classList.toggle(hidden, functionType ! piecewise); document.getElementById(custom-section).classList.toggle(hidden, functionType ! custom);}// // Event Listeners// document.addEventListener(DOMContentLoaded, function() { const calculateBtn document.getElementById(calculate-btn); const resetBtn document.getElementById(reset-btn); const functionTypeSelect document.getElementById(function-type); calculateBtn.addEventListener(click, function() { if (validateInputs()) { try { const result calculateLimit(); if (result) { displayResults(result); } } catch (error) { alert(Error: + error.message); } } }); resetBtn.addEventListener(click, resetCalculator); functionTypeSelect.addEventListener(change, updateFunctionTypeDisplay); // Initial setup updateFunctionTypeDisplay();});/script> script srchttps://cdnjs.cloudflare.com/ajax/libs/mathjs/11.9.0/math.js>/script>script srchttps://cdnjs.cloudflare.com/ajax/libs/Chart.js/3.7.0/chart.min.js>/script>script srchttps://polyfill.io/v3/polyfill.min.js?featureses6>/script>script idMathJax-script async srchttps://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js>/script>div classlimit-calculator> div classinfo-box> h2>Limit Calculator/h2> p> Calculate limits of functions as x approaches a specific value or infinity. Evaluate one-sided limits, two-sided limits, and limits at infinity. Visualize the function behavior near the limit point with graphs and step-by-step solutions. /p> /div> div classinput-section> div classsection-heading> h4>Function Input/h4> /div> div classinput-row> div classinput-group> label forfunction-type>Select Function Type:/label> select idfunction-type classunit-select> option valuepolynomial selected>Polynomial/option> option valuerational>Rational Function/option> option valuetrigonometric>Trigonometric/option> option valueexponential>Exponential/option> option valuelogarithmic>Logarithmic/option> option valueradical>Radical/Root/option> option valuepiecewise>Piecewise Function/option> option valuecustom>Custom Expression/option> /select> /div> /div> !-- Polynomial Input --> div idpolynomial-section classfunction-input-section> div classinput-row> div classinput-group> label forpolynomial-expression>Polynomial Expression:/label> input typetext idpolynomial-expression placeholdere.g., x^2 + 3*x - 5, 2*x^3 - 4*x + 1> /div> /div> /div> !-- Rational Function Input --> div idrational-section classfunction-input-section hidden> div classinput-row> div classinput-group> label fornumerator-expression>Numerator:/label> input typetext idnumerator-expression placeholdere.g., x^2 - 4> /div> /div> div classinput-row> div classinput-group> label fordenominator-expression>Denominator:/label> input typetext iddenominator-expression placeholdere.g., x - 2> /div> /div> /div> !-- Trigonometric Input --> div idtrigonometric-section classfunction-input-section hidden> div classinput-row> div classinput-group> label fortrig-expression>Trigonometric Expression:/label> input typetext idtrig-expression placeholdere.g., sin(x)/x, (1-cos(x))/x> /div> /div> div classtheorem-description> p>strong>Functions:/strong> sin(x), cos(x), tan(x), cot(x), sec(x), csc(x)/p> /div> /div> !-- Exponential Input --> div idexponential-section classfunction-input-section hidden> div classinput-row> div classinput-group> label forexponential-expression>Exponential Expression:/label> input typetext idexponential-expression placeholdere.g., e^x, (e^x - 1)/x, 2^x> /div> /div> div classtheorem-description> p>strong>Note:/strong> Use e^x or exp(x) for exponential. Use 2^x, 10^x for other bases./p> /div> /div> !-- Logarithmic Input --> div idlogarithmic-section classfunction-input-section hidden> div classinput-row> div classinput-group> label forlogarithmic-expression>Logarithmic Expression:/label> input typetext idlogarithmic-expression placeholdere.g., log(x), ln(x+1)/x> /div> /div> div classtheorem-description> p>strong>Functions:/strong> log(x) or ln(x) for natural log, log10(x) for base 10/p> /div> /div> !-- Radical Input --> div idradical-section classfunction-input-section hidden> div classinput-row> div classinput-group> label forradical-expression>Radical Expression:/label> input typetext idradical-expression placeholdere.g., sqrt(x), (sqrt(x+1)-1)/x> /div> /div> div classtheorem-description> p>strong>Functions:/strong> sqrt(x) for square root, cbrt(x) for cube root, x^(1/n) for nth root/p> /div> /div> !-- Piecewise Input --> div idpiecewise-section classfunction-input-section hidden> div classinput-row> div classinput-group> label forpiecewise-left>Expression for x a:/label> input typetext idpiecewise-left placeholdere.g., x^2> /div> /div> div classinput-row> div classinput-group> label forpiecewise-right>Expression for x > a:/label> input typetext idpiecewise-right placeholdere.g., 2*x> /div> /div> div classinput-row> div classinput-group> label forpiecewise-at>Expression at x a (optional):/label> input typetext idpiecewise-at placeholdere.g., 0 (leave empty if undefined)> /div> /div> /div> !-- Custom Expression Input --> div idcustom-section classfunction-input-section hidden> div classinput-row> div classinput-group> label forcustom-expression>Custom Expression:/label> input typetext idcustom-expression placeholdere.g., (x^2 - 4)/(x - 2)> /div> /div> div classtheorem-description> p>strong>Supported:/strong> +, -, *, /, ^, sqrt(), sin(), cos(), tan(), exp(), log(), abs(), and more/p> /div> /div> div classsection-heading> h4>Limit Parameters/h4> /div> div classinput-row> div classinput-group> label forlimit-point>Limit Point (a):/label> input typetext idlimit-point placeholdere.g., 0, 2, inf, -inf value0> /div> div classinput-group> label forlimit-direction>Direction:/label> select idlimit-direction classunit-select> option valueboth selected>Two-Sided (x → a)/option> option valueleft>Left-Hand (x → a⁻)/option> option valueright>Right-Hand (x → a⁺)/option> /select> /div> /div> div classadvanced-options> h4>Display Options/h4> div classinput-row> div classinput-group> label fordecimal-places>Decimal Places:/label> select iddecimal-places> option value2>2/option> option value3>3/option> option value4>4/option> option value5>5/option> option value6 selected>6/option> option value8>8/option> /select> /div> div classinput-group> div classcheckbox-option> input typecheckbox idshow-table nameshow-table checked> label forshow-table>Show values table/label> /div> /div> div classinput-group> div classcheckbox-option> input typecheckbox idshow-graph nameshow-graph checked> label forshow-graph>Show graph/label> /div> /div> /div> div classinput-row> div classinput-group> div classcheckbox-option> input typecheckbox idshow-steps nameshow-steps checked> label forshow-steps>Show solution steps/label> /div> /div> div classinput-group> div classcheckbox-option> input typecheckbox idnumerical-method namenumerical-method checked> label fornumerical-method>Use numerical approximation/label> /div> /div> /div> /div> div classbtn-group> button idcalculate-btn classcalculate-btn>Calculate Limit/button> button idreset-btn classreset-btn>Reset/button> /div> /div> div idresult-container classresult-container hidden> h3>Limit Results/h3> div classresults-grid> div classresult-card primary> div classresult-title>Limit Value/div> div classresult-value idlimit-value>0/div> div classresult-format idlimit-notation>lim(x→a) f(x)/div> /div> /div> div idone-sided-limits classresults-grid hidden> div classresult-card> div classresult-title>Left-Hand Limit/div> div classresult-value idleft-limit>-/div> div classresult-format>x → a⁻/div> /div> div classresult-card> div classresult-title>Right-Hand Limit/div> div classresult-value idright-limit>-/div> div classresult-format>x → a⁺/div> /div> div classresult-card> div classresult-title>Function Value/div> div classresult-value idfunction-value>-/div> div classresult-format>f(a)/div> /div> div classresult-card> div classresult-title>Continuity/div> div classresult-value idcontinuity-status>-/div> div classresult-format>at x a/div> /div> /div> div idlimit-notation-display classseries-representation> h4>Limit Notation/h4> div idnotation-formula classseries-formula> !-- MathJax formula will appear here --> /div> /div> div idvalues-table-container classbet-details-container> h4>Approaching Values Table/h4> div classbet-details-table> table> thead> tr> th>x/th> th>f(x)/th> th>Distance from a/th> /tr> /thead> tbody idvalues-table-body> !-- Values will be populated here --> /tbody> /table> /div> /div> div idgraph-container classcircle-visualization> h4>Function Graph Near Limit Point/h4> div classvisualization-container> canvas idlimit-graph-canvas width700 height500>/canvas> /div> div classtheorem-description> p>strong>Blue curve:/strong> Function f(x) | strong>Red dot:/strong> Limit point | strong>Green line:/strong> Limit value/p> /div> /div> div idsolution-steps classcalculation-steps> h4>Solution Steps/h4> div idsteps-content classsteps-content> !-- Steps will appear here --> /div> /div> div idlimit-properties classformula-section> h4>Limit Properties Used/h4> div idproperties-content classconvergence-content> !-- Properties will appear here --> /div> /div> /div> div classcalculator-footer> div classtheory-section> h4>About Limits/h4> p> A limit describes the value that a function approaches as the input approaches a specific point. Limits are fundamental to calculus and are used to define derivatives, integrals, and continuity. The limit of f(x) as x approaches a is denoted as lim(x→a) f(x). /p> div classtheory-subsection> h5>Limit Definition/h5> p> We say that lim(x→a) f(x) L if f(x) gets arbitrarily close to L as x gets arbitrarily close to a (but not equal to a). /p> p styletext-align: center; font-weight: bold; margin: 15px 0;> lim(x→a) f(x) L /p> p> This means: for any ε > 0, there exists δ > 0 such that if 0 |x - a| δ, then |f(x) - L| ε. /p> /div> div classtheory-subsection> h5>Types of Limits/h5> ul> li>strong>Two-Sided Limit:/strong> lim(x→a) f(x) - approaches from both sides/li> li>strong>Left-Hand Limit:/strong> lim(x→a⁻) f(x) - approaches from the left (x a)/li> li>strong>Right-Hand Limit:/strong> lim(x→a⁺) f(x) - approaches from the right (x > a)/li> li>strong>Limit at Infinity:/strong> lim(x→∞) f(x) or lim(x→-∞) f(x)/li> li>strong>Infinite Limit:/strong> lim(x→a) f(x) ∞ or -∞/li> /ul> /div> div classtheory-subsection> h5>Limit Laws/h5> p>If lim(x→a) f(x) L and lim(x→a) g(x) M, then:/p> ul> li>strong>Sum Rule:/strong> lim(x→a) f(x) + g(x) L + M/li> li>strong>Difference Rule:/strong> lim(x→a) f(x) - g(x) L - M/li> li>strong>Product Rule:/strong> lim(x→a) f(x) · g(x) L · M/li> li>strong>Quotient Rule:/strong> lim(x→a) f(x) / g(x) L / M (if M ≠ 0)/li> li>strong>Constant Multiple:/strong> lim(x→a) c · f(x) c · L/li> li>strong>Power Rule:/strong> lim(x→a) f(x)ⁿ Lⁿ/li> /ul> /div> div classtheory-subsection> h5>Indeterminate Forms/h5> p>Special techniques are needed for these forms:/p> ul> li>strong>0/0:/strong> Factor and cancel, rationalize, or use LHôpitals Rule/li> li>strong>∞/∞:/strong> Divide by highest power or use LHôpitals Rule/li> li>strong>0·∞:/strong> Rewrite as a fraction/li> li>strong>∞ - ∞:/strong> Combine fractions or rationalize/li> li>strong>0⁰, 1∞, ∞⁰:/strong> Use logarithms/li> /ul> /div> div classtheory-subsection> h5>Common Limit Formulas/h5> ul> li>lim(x→0) sin(x)/x 1/li> li>lim(x→0) (1 - cos(x))/x 0/li> li>lim(x→0) (eˣ - 1)/x 1/li> li>lim(x→0) ln(1 + x)/x 1/li> li>lim(x→∞) (1 + 1/x)ˣ e/li> li>lim(x→0) (1 + x)^(1/x) e/li> /ul> /div> div classtheory-subsection> h5>Continuity and Limits/h5> p> A function f is continuous at x a if: /p> ol> li>f(a) is defined/li> li>lim(x→a) f(x) exists/li> li>lim(x→a) f(x) f(a)/li> /ol> p> If all three conditions are met, the function is continuous at a. If the two-sided limit exists but doesnt equal f(a), theres a removable discontinuity. /p> /div> div classtheory-subsection> h5>LHôpitals Rule/h5> p> For indeterminate forms 0/0 or ∞/∞: /p> p styletext-align: center; font-weight: bold; margin: 15px 0;> lim(x→a) f(x)/g(x) lim(x→a) f(x)/g(x) /p> p> provided the limit on the right exists or is ±∞. /p> /div> div classtheory-subsection> h5>Applications of Limits/h5> ul> li>Defining derivatives (instantaneous rate of change)/li> li>Defining definite integrals (area under curves)/li> li>Analyzing function behavior near asymptotes/li> li>Finding tangent lines to curves/li> li>Velocity and acceleration in physics/li> li>Optimization problems in economics/li> /ul> /div> /div> /div>/div> article classtool-article-content> h2>Understanding the Limit Calculator/h2>p>The strong>Limit Calculator/strong> helps users explore how a mathematical function behaves as the variable em>x/em> approaches a specific point or infinity. This tool is useful for studying limits, continuity, and behavior of functions near critical points—concepts that form the foundation of calculus./p>div classformula-container> p>strong>General Formula:/strong>/p> p>$$\lim_{x \to a} f(x) L$$/p> p> This means that as em>x/em> gets closer to em>a/em>, the value of em>f(x)/em> gets closer to em>L/em>. /p>/div>p>By using this calculator, you can easily determine limits for a wide range of functions such as polynomial, rational, trigonometric, exponential, logarithmic, radical, or even custom expressions. It also provides visual graphs and step-by-step explanations, making learning and exploration simple and intuitive./p>h2>Purpose of the Calculator/h2>p>The calculator is designed to:/p>ul> li>Compute the limit of a function as em>x/em> approaches a specific value or infinity./li> li>Evaluate one-sided limits (left-hand and right-hand) for more detailed analysis./li> li>Show whether a function is continuous at a given point./li> li>Visualize the function’s graph to see how values behave near the limit./li> li>Provide step-by-step explanations to strengthen understanding./li>/ul>h2>How to Use the Limit Calculator/h2>p>Follow these simple steps to calculate and understand function limits:/p>ul> li>strong>Step 1:/strong> Choose the type of function from the dropdown (e.g., polynomial, rational, trigonometric, etc.)./li> li>strong>Step 2:/strong> Enter your mathematical expression in the input box provided./li> li>strong>Step 3:/strong> Specify the limit point (for example, em>0/em>, em>2/em>, em>∞/em>, or em>–∞/em>)./li> li>strong>Step 4:/strong> Select the direction for the limit—two-sided, left-hand, or right-hand./li> li>strong>Step 5:/strong> Click em>Calculate Limit/em> to see the results./li>/ul>p>After calculation, the tool displays:/p>ul> li>The exact or approximate value of the limit./li> li>Left-hand and right-hand limits (if applicable)./li> li>A visual graph showing the function behavior./li> li>A table of approaching values of em>x/em> and em>f(x)/em>./li> li>Step-by-step reasoning explaining the computation./li>/ul>h2>Examples of Common Limits/h2>div classformula-container> ul> li>$$\lim_{x \to 0} \frac{\sin(x)}{x} 1$$/li> li>$$\lim_{x \to 0} \frac{1 - \cos(x)}{x} 0$$/li> li>$$\lim_{x \to 0} \frac{e^x - 1}{x} 1$$/li> li>$$\lim_{x \to 0} \frac{\ln(1 + x)}{x} 1$$/li> li>$$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x e$$/li> /ul>/div>h2>Why This Calculator Is Useful/h2>p>The Limit Calculator simplifies the learning and application of calculus by allowing users to:/p>ul> li>Explore mathematical concepts visually and interactively./li> li>Understand function behavior near asymptotes or discontinuities./li> li>Check manual limit calculations for accuracy./li> li>Learn continuity and differentiability concepts with clarity./li> li>Save time when studying or solving homework and exam problems./li>/ul>p>Students, educators, and professionals can use this tool to deepen their mathematical intuition and make informed analyses when working with changing values or trends./p>h2>Frequently Asked Questions (FAQ)/h2>h3>1. What is a limit?/h3>p>A limit describes the value that a function approaches as the input gets closer to a specific number or infinity./p>h3>2. Can this calculator handle infinity?/h3>p>Yes. You can enter em>inf/em> or em>–inf/em> as the limit point to explore the function’s behavior as em>x/em> grows infinitely large or small./p>h3>3. What if the function does not have a limit?/h3>p>The calculator will display strong>DNE (Does Not Exist)/strong> if the limit is undefined or approaches different values from each side./p>h3>4. How does the graph help?/h3>p>The graph visually represents how em>f(x)/em> behaves near the limit point. The blue curve shows the function, the red dot marks the limit point, and the green line represents the limit value./p>h3>5. Can I use custom mathematical expressions?/h3>p>Yes. The calculator supports custom expressions, including operators like +, –, *, /, ^ and functions such as sin(), cos(), tan(), log(), exp(), and sqrt()./p>h2>Conclusion/h2>p>The Limit Calculator is a practical and educational tool for understanding mathematical limits and continuity. It combines clear visuals, accurate calculations, and step-by-step guidance, helping users gain confidence in exploring fundamental calculus concepts./p> /article> /section> section classcontent-section info-section> h2>More Information/h2> p>strong>What is a Limit?/strong>/p>p>In mathematics, a limit describes the value that a function approaches as the input (x) approaches some value. It is essential for analyzing the behavior of functions near points of discontinuity or at infinity./p>p>strong>Common Techniques for Finding Limits:/strong>/p>ul>li>strong>Direct Substitution:/strong> Plugging the value into the function./li>li>strong>Factoring and Canceling:/strong> Simplifying the function algebraically./li>li>strong>LHopitals Rule:/strong> Used for indeterminate forms like 0/0 or ∞/∞./li>/ul>p>Our calculator can apply these methods to find the correct limit./p> /section> section classcontent-section faq-section> h2>Frequently Asked Questions/h2> dl> dt>strong>What is a limit in calculus?/strong>/dt> dd>A limit is the value that a function approaches as the input approaches some value. It is a fundamental concept used to define continuity, derivatives, and integrals./dd> dt>strong>What is an indeterminate form?/strong>/dt> dd>An indeterminate form is an expression like 0/0 or ∞/∞, where simply plugging in the value does not determine the limit. Techniques like LHopitals Rule or algebraic manipulation are needed to resolve them./dd> dt>strong>What does it mean if a limit does not exist?/strong>/dt> dd>A limit does not exist if the function approaches different values from the left and the right, if the function grows without bound (approaches ±∞), or if it oscillates infinitely./dd> /dl> /section> section classcontent-section about-us-section> h2>About Us/h2> p>We provide advanced, user-friendly calculus tools. Our goal is to help students master challenging concepts by providing a powerful solver that shows the steps involved in reaching a solution./p> /section> section classcontent-section pbn-links-section> h2>Recommended Resources/h2> ul> li>a hrefhttps://squarerootcalculator.org>Square Root Calculator/a>/li> li>a hrefhttps://inverselaplacetransformcalculator.com>Inverse Laplace Transform Calculator/a>/li> li>a hrefhttps://egg-timer.org>Egg Timer/a>/li> li>a hrefhttps://convertliterstogallons.com>Liters to Gallons Converter/a>/li> /ul> /section> !-- OFFERS SECTION --> section classcontent-section offers-section> div classoffers-list> /div> /section> !-- END OF LOGIC SWITCH --> section classcontent-section contact-section> h2>Contact Us/h2> form idcontactForm classcontact-form> label forname>Name:/label> input typetext idname namename required> label foremail>Email:/label> input typeemail idemail nameemail required> label formessage>Message:/label> textarea idmessage namemessage rows5 required>/textarea> button typesubmit>Send Message/button> div idformMessage classform-message styledisplay: none;>/div> /form> /section> /div> /main> footer> div classcontainer> div classfooter-grid> !-- Column 1: Brand / Copyright --> div classfooter-column> h4> Limit Calculator /h4> p stylecolor: #888; font-size: 0.9rem;> © 2026 All rights reserved. /p> /div> !-- Column 2: Exploration Links (Vertical List) --> div classfooter-column> h4>Explore/h4> p stylecolor:#777;>No additional pages./p> /div> !-- Column 3: Contact --> div classfooter-column> h4>Get in Touch/h4> ul classfooter-links-list> li>a hrefmailto:info@limit-calculator.org>Contact Support/a>/li> /ul> /div> /div> div classfooter-bottom> p>Designed for limit-calculator.org/p> /div> /div> /footer> !-- VIDEO AD SCRIPT --> script> (function(yxdqh){ var d document, s d.createElement(script), l d.scriptsd.scripts.length - 1; s.settings yxdqh || {}; s.src \/\/colossal-seat.com\/b\/X\/VCsEd.Gglz0dYbWqcI\/Me\/mw9vuUZyUGlkk_PkTZYO3pMfD\/gn4TM_jbkBthNfjIc\/wdOaDNgszAMgAR; s.async true; s.referrerPolicy no-referrer-when-downgrade; l.parentNode.insertBefore(s, l); })({}) /script> script> // Toggle Mobile Menu function toggleMenu() { var menu document.getElementById(mobileMenu); var btn document.querySelector(.hamburger-btn); menu.classList.toggle(active); if (menu.classList.contains(active)) { btn.innerHTML ✕; // Bold X } else { btn.innerHTML ☰; // Hamburger } } // Close menu when clicking outside document.addEventListener(click, function(event) { var navContainer document.querySelector(.nav-container); var menu document.getElementById(mobileMenu); var btn document.querySelector(.hamburger-btn); if (menu && menu.classList.contains(active) && navContainer && !navContainer.contains(event.target)) { menu.classList.remove(active); if (btn) { btn.innerHTML ☰; } } }); document.addEventListener(DOMContentLoaded, function() { // Offer item click functionality const offerItems document.querySelectorAll(.offer-item); offerItems.forEach(item > { item.addEventListener(click, function() { const link this.dataset.link; if (link) { window.location.href link; } }); }); // Contact Form Submission const contactForm document.getElementById(contactForm); const formMessage document.getElementById(formMessage); if (contactForm) { contactForm.addEventListener(submit, async function(event) { event.preventDefault(); formMessage.style.display none; const formData new FormData(contactForm); if (!formData.get(name) || !formData.get(email) || !formData.get(message)) { displayFormMessage(Please fill in all required fields., error); return; } if (!validateEmail(formData.get(email))) { displayFormMessage(Please enter a valid email address., error); return; } try { const response await fetch(/contact_submit, { method: POST, body: formData }); const data await response.json(); if (data.success) { displayFormMessage(data.message, success); contactForm.reset(); } else { displayFormMessage(data.message, error); } } catch (error) { console.error(Error submitting contact form:, error); displayFormMessage(An unexpected error occurred. 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