Visualize and understand the behavior of functions as they approach specific points.
Interactive Limits Calculator
Enter a function of x (e.g., 'x^2 + 2x – 1', 'sin(x)/x'). Use standard math notation.
The value x approaches. Use 'Infinity' or '-Infinity' for limits at infinity.
Both (+/-)
From the right (+)
From the left (-)
Specify if approaching from the positive or negative side, or both.
Number of sample points to approximate the limit. Higher values give better accuracy but may take longer.
Limit Analysis
—
Limit (L): —
Left-Hand Limit (L⁻): —
Right-Hand Limit (L⁺): —
Behavior near 'a': —
Formula & Method: This calculator approximates the limit of a function f(x) as x approaches a point a. It does this by evaluating the function at points very close to a (from both sides, if specified). If the function values converge to a single number L, that is the limit. For limits at infinity, it samples points with large magnitudes.
Function Graph Visualization
Graph showing function f(x) near point 'a'.
Limit Calculation Summary
Parameter
Value
Unit
Notes
Function
—
N/A
The function being analyzed.
Approaching Point (a)
—
X-unit
The value x tends towards.
Direction
—
N/A
Direction of approach to 'a'.
Approximation Points
—
Count
Number of sample points used.
Calculated Limit (L)
—
Y-unit
The estimated limit value.
What is Limits Graphing Calculator?
A Limits Graphing Calculator is an essential tool in calculus designed to help visualize and compute the limit of a function as the input variable approaches a specific value or infinity. It goes beyond simple numerical calculation by often integrating graphical representations, allowing users to see how the function's output behaves near a target point. Understanding limits is fundamental to calculus, underpinning concepts like continuity, derivatives (rates of change), and integrals (areas under curves).
Who should use it:
Students: High school and college students learning calculus, pre-calculus, or analysis.
Educators: Teachers and professors demonstrating limit concepts visually and interactively.
Engineers & Scientists: Professionals who need to analyze function behavior at critical points or in asymptotic scenarios.
Mathematicians: Researchers and enthusiasts exploring function properties.
Common Misconceptions:
Limit equals function value: A common mistake is assuming that the limit of a function at a point must be equal to the function's value *at* that point. While this is true for continuous functions, limits describe behavior *near* a point, not necessarily *at* it. A function can have a limit at a point where it's undefined (e.g., a hole in the graph).
Limit always exists: Limits don't always exist. They might fail to exist if the function oscillates infinitely, approaches different values from the left and right, or grows without bound (approaches infinity).
Limit means plugging in the value: While direct substitution works for many well-behaved functions (continuous functions), it fails for indeterminate forms like 0/0 or ∞/∞, where more sophisticated methods (like algebraic manipulation, L'Hôpital's Rule, or numerical approximation) are required.
Limits Graphing Calculator Formula and Mathematical Explanation
The core concept of a limit is to understand the value a function f(x) tends toward as its input x gets arbitrarily close to a specific value a, without necessarily reaching a. Mathematically, we write this as:
lim┬(x→a)〖f(x)〗 = L
This states that the limit of f(x) as x approaches a is L.
Numerical Approximation Method:
Our Limits Graphing Calculator uses a numerical approximation method. Instead of relying solely on symbolic manipulation (which can be complex for arbitrary functions), it samples the function at numerous points very close to 'a'.
Define Sample Points: Based on the desired 'Precision' (let's call it N), we generate N points slightly less than a (x⁻ᵢ) and N points slightly greater than a (x⁺ᵢ). The spacing between these points decreases as they approach a. If 'Infinity' or '-Infinity' is the point 'a', points with large absolute values are used.
Evaluate Function: Calculate f(x⁻ᵢ) and f(x⁺ᵢ) for all sample points.
Analyze Convergence:
Left-Hand Limit (L⁻): Observe the values of f(x⁻ᵢ) as x⁻ᵢ gets closer to a. If these values cluster around a specific number, that number is the left-hand limit.
Right-Hand Limit (L⁺): Observe the values of f(x⁺ᵢ) as x⁺ᵢ gets closer to a. If these values cluster around a specific number, that number is the right-hand limit.
Two-Sided Limit (L): The overall limit L exists if and only if the left-hand limit equals the right-hand limit (L⁻ = L⁺). In this case, L = L⁻ = L⁺.
Handle Special Cases:
Infinity/Negative Infinity: If 'a' is Infinity or Negative Infinity, the calculator evaluates f(x) for very large positive or negative x values.
Undefined Points: If f(a) results in an error (like division by zero), the calculator relies entirely on the behavior of points *around* 'a'.
Variables Table:
Variable
Meaning
Unit
Typical Range/Type
f(x)
The function being analyzed.
Depends on function
Mathematical expression of x.
x
The independent input variable.
X-unit
Real number or Infinity.
a
The point x approaches.
X-unit
Real number, Infinity, or -Infinity.
L
The limit value of f(x) as x approaches a.
Y-unit
Real number or Infinity.
L⁻
The left-hand limit (x approaches a from values < a).
Y-unit
Real number or Infinity.
L⁺
The right-hand limit (x approaches a from values > a).
Y-unit
Real number or Infinity.
N
Number of sample points (Precision).
Count
Integer >= 10.
Practical Examples (Real-World Use Cases)
Example 1: Hole in a Rational Function
Consider the function f(x) = (x^2 - 1) / (x - 1). We want to find the limit as x approaches 1.
Input:
Function f(x): (x^2 - 1) / (x - 1)
Point 'a': 1
Direction: Both (+/-)
Precision: 100
Calculator Output:
Primary Result: 2
Limit (L): 2
Left-Hand Limit (L⁻): 2
Right-Hand Limit (L⁺): 2
Behavior near 'a': The function approaches 2.
Interpretation: If you plug x=1 directly into the function, you get 0/0, an indeterminate form. However, by simplifying the function algebraically (f(x) = (x-1)(x+1) / (x-1) = x+1 for x ≠ 1), we see it behaves like the line y = x + 1 everywhere except at x=1, where there's a "hole". The limit correctly identifies the y-value (2) that the function approaches near x=1.
Example 2: Limit at Infinity (Horizontal Asymptote)
Consider the function f(x) = (3x^2 + 5) / (x^2 - 2). We want to find the limit as x approaches positive infinity.
Input:
Function f(x): (3x^2 + 5) / (x^2 - 2)
Point 'a': Infinity
Direction: Both (+/-) (relevant for infinity)
Precision: 200
Calculator Output:
Primary Result: 3
Limit (L): 3
Left-Hand Limit (L⁻): 3 (Approaching from negative infinity)
Right-Hand Limit (L⁺): 3 (Approaching from positive infinity)
Behavior near 'a': The function approaches 3.
Interpretation: As x becomes very large (positive or negative), the constant terms (+5 and -2) become insignificant compared to the x² terms. The function behaves like 3x²/x², which simplifies to 3. The calculator numerically confirms that the function has a horizontal asymptote at y = 3. This is crucial for understanding the long-term behavior of the function.
Example 3: Limit Does Not Exist (Jump Discontinuity)
Consider a piecewise function: f(x) = 1 if x < 0, and f(x) = 2 if x >= 0. We want the limit as x approaches 0.
Input:
Function f(x): Piecewise(1 if x=0) *(Note: This calculator might not directly support piecewise input syntax like this. For demonstration, imagine we input the parts separately or use a system that understands it. Our current calculator is best for single expressions.)* Let's simulate by focusing on the *concept*. If we tried to graph `y=1` for x=0, we'd see a jump.
Point 'a': 0
Direction: Both (+/-)
Precision: 100
Calculator Output (Conceptual):
Primary Result: DNE
Limit (L): DNE
Left-Hand Limit (L⁻): 1
Right-Hand Limit (L⁺): 2
Behavior near 'a': The function approaches different values from the left and right.
Interpretation: Since L⁻ (1) ≠ L⁺ (2), the overall two-sided limit does not exist at x=0. This indicates a jump discontinuity. The calculator's ability to distinguish left and right limits is key here.
How to Use This Limits Graphing Calculator
Enter the Function: In the "Function f(x)" field, type the mathematical expression for the function you want to analyze. Use standard notation like x^2 for x squared, sin(x), cos(x), exp(x) or e^x for exponential, etc. For functions involving infinity, you might use specific syntax if supported, or rely on the 'Point' input.
Specify the Point: Enter the value 'a' that you want 'x' to approach in the "Point 'a' to approach" field. You can enter a specific number (like 0, 2, -5) or use terms like 'Infinity' or '-Infinity' if your input method supports it (our calculator uses numerical approximation, so large numbers represent infinity).
Choose Approach Direction: Select "Both (+/-)" if you want to find the standard two-sided limit. Choose "From the right (+)" to calculate the limit as x approaches 'a' from values greater than 'a' (L⁺). Choose "From the left (-)" for the limit as x approaches 'a' from values less than 'a' (L⁻).
Set Precision: Adjust the "Calculation Precision" slider. A higher number means the calculator will check more points very close to 'a', leading to a more accurate approximation, especially for complex functions. Start with the default (e.g., 100) and increase if needed.
Calculate: Click the "Calculate Limit" button.
Interpret Results:
Primary Result: This shows the main computed limit value (L) or 'DNE' (Does Not Exist) if the left and right limits differ or if the function diverges.
Limit (L): The overall limit value.
Left-Hand Limit (L⁻) / Right-Hand Limit (L⁺): The values the function approaches from each side. If these are different, the main limit (L) Does Not Exist.
Behavior near 'a': A brief text summary of the function's behavior.
Graph: The visualization helps you see the function's trend around 'a'.
Table: Provides a summary of the inputs and key outputs.
Decision Guidance: If L⁻ = L⁺, the limit exists and the function is likely continuous or has a removable discontinuity (hole) at 'a'. If L⁻ ≠ L⁺, there's a jump discontinuity, and the limit DNE. If the function values grow infinitely large (positive or negative), the limit is Infinity, -Infinity, or DNE. Understanding these helps determine function continuity, identify asymptotes, and analyze stability in systems.
Reset: Click "Reset" to clear all inputs and results and return to default settings.
Copy Results: Click "Copy Results" to copy the primary result, intermediate values, and key assumptions to your clipboard for documentation or sharing.
Key Factors That Affect Limits Results
Several factors influence the calculation and interpretation of limits:
Function Definition: The algebraic structure of f(x) is paramount. Polynomials are generally continuous, rational functions can have holes or vertical asymptotes, and trigonometric or exponential functions have unique behaviors. The specific expression dictates whether the limit exists and what its value might be.
The Point 'a': Limits are evaluated *as x approaches a*. Whether 'a' is a finite number, zero, infinity, or negative infinity drastically changes the analysis. Limits at finite points often relate to continuity and local behavior, while limits at infinity describe end behavior and asymptotes.
Continuity of the Function: For continuous functions at point 'a', the limit is simply the function's value: lim┬(x→a)〖f(x)〗 = f(a). The complexity arises with discontinuities (removable, jump, infinite). This calculator helps identify these by comparing numerical approximations.
Indeterminate Forms (0/0, ∞/∞): When direct substitution yields these forms, it means more analysis is needed. Algebraic simplification (factoring, rationalizing), L'Hôpital's Rule (requiring derivatives), or the numerical method used here are necessary to find the true limit. Our calculator excels at providing numerical insight in these cases.
Rate of Convergence: The 'Precision' setting relates to how quickly the function values approach the limit. A function that converges rapidly will show a stable limit even with fewer precision points, while a slowly converging function might require higher precision for an accurate approximation. This relates to the function's behavior near 'a' – is it steep, shallow, or oscillating?
Computational Precision & Errors: Floating-point arithmetic in computers has limitations. For extremely complex functions or points very close to singularities, numerical methods might introduce small rounding errors. The calculator's precision setting helps mitigate this by using more sample points.
Frequently Asked Questions (FAQ)
Q1: What's the difference between a limit and a function value f(a)?A limit describes the value a function *approaches* as x gets close to 'a', while f(a) is the actual value *at* x=a. They are often the same for continuous functions, but a limit can exist where f(a) is undefined (a hole) or different from the limit (a removable discontinuity).
Q2: When does a limit not exist (DNE)?A limit DNE if: 1) The left-hand limit (L⁻) is not equal to the right-hand limit (L⁺). 2) The function grows infinitely large (approaches ±∞) without settling on a specific value. 3) The function oscillates infinitely without approaching any single value.
Q3: Can I use this calculator for limits involving infinity?Yes, by entering 'Infinity' or '-Infinity' (or very large numbers) in the 'Point 'a' to approach' field. The calculator will sample points with large magnitudes to approximate the end behavior and horizontal asymptotes.
Q4: How accurate is the numerical approximation?The accuracy depends on the 'Calculation Precision' setting and the function's behavior. Higher precision samples more points, improving accuracy, especially for complex or slowly converging functions. However, extreme precision can be computationally intensive and encounter floating-point limitations.
Q5: What if my function involves piecewise definitions?Our calculator is designed primarily for single-expression functions. For piecewise functions, you would typically analyze each piece separately around the critical points or use a calculator specifically designed for piecewise analysis. The core concepts of left/right limits still apply.
Q6: Does the calculator use L'Hôpital's Rule?This specific calculator uses numerical approximation by sampling points. It does not perform symbolic differentiation required for L'Hôpital's Rule, but it can provide results consistent with L'Hôpital's Rule by observing the trend of function values near the point.
Q7: What does "Behavior near 'a'" mean?This provides a concise summary, such as "Approaches a finite value L", "Approaches positive infinity", "Approaches negative infinity", or "Approaches different values from left and right", helping to interpret the numerical limit results.
Q8: How can I verify the calculator's results?Compare the results with algebraic methods (simplification, factoring), graphical analysis using graphing software, or by manually checking function values at points extremely close to 'a'. Understanding the underlying calculus concepts is key to verification.
Understand vertical, horizontal, and slant asymptotes, often determined using limits.
var chartInstance = null; // Store chart instance globally
function getInputElement(id) {
return document.getElementById(id);
}
function getElement(id) {
return document.getElementById(id);
}
function showError(elementId, message) {
var errorElement = getElement(elementId);
if (errorElement) {
errorElement.textContent = message;
}
}
function clearError(elementId) {
showError(elementId, ");
}
function isValidNumber(value) {
return !isNaN(parseFloat(value)) && isFinite(value);
}
function parseFunction(funcStr, xValue) {
try {
if (xValue === Infinity) {
xValue = 1e30; // Use a very large number for Infinity approximation
} else if (xValue === -Infinity) {
xValue = -1e30; // Use a very large negative number
}
// Basic safety against extremely large/small numbers that might cause overflow
if (Math.abs(xValue) > 1e100) {
return NaN; // Indicate overflow/too large
}
// Replace math functions and constants
funcStr = funcStr.replace(/pi/gi, Math.PI.toString());
funcStr = funcStr.replace(/e\^/gi, 'Math.exp(');
funcStr = funcStr.replace(/\^/gi, '**'); // Use exponentiation operator
// Ensure x is available in the scope for evaluation
var x = xValue;
var result = math.evaluate(funcStr, { x: x });
// Check for potential errors during evaluation
if (typeof result === 'number' && !isFinite(result)) {
return result; // Return Infinity or -Infinity if the calculation results in it
}
if (typeof result !== 'number' || isNaN(result)) {
return NaN; // Indicates an error in the function expression or evaluation
}
return result;
} catch (e) {
// console.error("Error evaluating function:", e);
return NaN; // Indicates an error in the function expression
}
}
function calculateLimit() {
var functionInput = getInputElement('functionInput').value.trim();
var pointInput = getInputElement('pointInput').value.trim();
var direction = getElement('directionSelect').value;
var precision = parseInt(getElement('precisionInput').value);
// Clear previous errors and results
clearError('functionError');
clearError('pointError');
clearError('precisionError');
getElement('resultsContainer').style.display = 'none';
getElement('primaryResult').textContent = '–';
getElement('limitValue').innerHTML = 'Limit (L): —';
getElement('leftLimitValue').innerHTML = 'Left-Hand Limit (L⁻): —';
getElement('rightLimitValue').innerHTML = 'Right-Hand Limit (L⁺): —';
getElement('functionBehavior').innerHTML = 'Behavior near \'a\': —';
getElement('summaryFunc').textContent = '–';
getElement('summaryPoint').textContent = '–';
getElement('summaryDirection').textContent = '–';
getElement('summaryPrecision').textContent = '–';
getElement('summaryLimit').textContent = '–';
if (functionInput === ") {
showError('functionError', 'Please enter a function.');
return;
}
if (pointInput === ") {
showError('pointError', 'Please enter the point x approaches.');
return;
}
if (isNaN(precision) || precision 1000) {
showError('precisionError', 'Precision must be between 10 and 1000.');
return;
}
var a;
var isPointInfinity = false;
if (pointInput.toLowerCase() === 'infinity') {
a = Infinity;
isPointInfinity = true;
} else if (pointInput.toLowerCase() === '-infinity') {
a = -Infinity;
isPointInfinity = true;
} else if (!isValidNumber(pointInput)) {
showError('pointError', 'Invalid number or "Infinity" / "-Infinity" expected.');
return;
} else {
a = parseFloat(pointInput);
// Check for negative zero if needed, though usually handled by parseFloat
if (Object.is(a, -0)) a = 0;
}
// — Calculations —
var leftLimit = NaN;
var rightLimit = NaN;
var limit = NaN;
var behavior = ";
var limitStr = '–';
var pointsCount = precision;
var stepSizeFactor = 10; // Start with a small step
var maxSteps = 100; // Limit iterations to prevent infinite loops
// Generate points for approximation
var x_vals_left = [];
var y_vals_left = [];
var x_vals_right = [];
var y_vals_right = [];
if (isPointInfinity) {
// Points for Infinity
var base = 1e5; // Starting large value
var multiplier = 1.5; // Grow exponentially
for (var i = 0; i < pointsCount && x_vals_left.length 0) break;
}
}
} else {
// Points for finite 'a'
var initialStep = Math.max(1, Math.abs(a) / 1000) / stepSizeFactor; // Initial step size, avoid zero step
if (a === 0) initialStep = 1 / stepSizeFactor; // Special case for a=0
for (var i = 0; i < pointsCount && x_vals_left.length < maxSteps; i++) {
var step = initialStep / Math.pow(stepSizeFactor, i);
if (step 0) { // Stop if evaluation fails after first point
break;
}
}
if (direction === 'both' || direction === 'positive') {
var x_right = a + step;
var y_right = parseFunction(functionInput, x_right);
if (!isNaN(y_right)) {
x_vals_right.push(x_right);
y_vals_right.push(y_right);
} else if (i > 0) { // Stop if evaluation fails after first point
break;
}
}
// Break if no valid points are generated in an iteration
if ( ( (direction === 'both' || direction === 'negative') && x_vals_left.length === i) ||
( (direction === 'both' || direction === 'positive') && x_vals_right.length === i) ) {
if(i>0) break; // Allow first point to potentially fail
}
}
}
// Calculate average of last few points for limit estimation
var numAvgPoints = Math.min(5, pointsCount / 2, maxSteps / 2); // Average last 5 or fewer points
if (numAvgPoints 0 || x_vals_right.length > 0)) {
numAvgPoints = 1; // Ensure at least one point is used if available
}
if (x_vals_left.length >= numAvgPoints) {
var sumYLeft = 0;
for (var i = 0; i = numAvgPoints) {
var sumYRight = 0;
for (var i = 0; i < numAvgPoints; i++) {
sumYRight += y_vals_right[y_vals_right.length – 1 – i];
}
rightLimit = sumYRight / numAvgPoints;
}
// Determine overall limit and behavior
var tolerance = 1e-6; // Tolerance for comparing limits
if (isPointInfinity) {
limit = leftLimit; // For infinity, left/right are conceptually the same approach direction
if (!isNaN(limit)) {
limitStr = limit.toFixed(6); // Format to 6 decimal places
behavior = "The function approaches " + limit.toFixed(2) + " as x approaches " + (a === Infinity ? "infinity" : "negative infinity");
} else {
behavior = "The function does not appear to approach a finite value.";
}
} else {
// Check if both limits were calculated successfully
var leftValid = !isNaN(leftLimit) && isFinite(leftLimit);
var rightValid = !isNaN(rightLimit) && isFinite(rightLimit);
if (direction === 'both') {
if (leftValid && rightValid && Math.abs(leftLimit – rightLimit) < tolerance) {
limit = (leftLimit + rightLimit) / 2; // Average them for final limit
limitStr = limit.toFixed(6);
behavior = "The function approaches " + limit.toFixed(2) + " from both sides.";
} else if (leftValid && rightValid) {
behavior = "The function approaches different values from the left and right.";
limitStr = "DNE";
} else if (leftValid) {
behavior = "The function approaches " + leftLimit.toFixed(2) + " from the left, but behavior from the right is unclear or diverges.";
limitStr = "DNE";
} else if (rightValid) {
behavior = "The function approaches " + rightLimit.toFixed(2) + " from the right, but behavior from the left is unclear or diverges.";
limitStr = "DNE";
} else {
behavior = "Could not determine the limit. Function may diverge or be undefined.";
limitStr = "DNE";
}
} else if (direction === 'negative') {
limit = leftLimit;
if (leftValid) {
limitStr = limit.toFixed(6);
behavior = "The function approaches " + limit.toFixed(2) + " from the left.";
} else {
behavior = "Could not determine the left-hand limit. Function may diverge.";
limitStr = "DNE";
}
} else { // positive direction
limit = rightLimit;
if (rightValid) {
limitStr = limit.toFixed(6);
behavior = "The function approaches " + limit.toFixed(2) + " from the right.";
} else {
behavior = "Could not determine the right-hand limit. Function may diverge.";
limitStr = "DNE";
}
}
}
// Update UI
getElement('primaryResult').textContent = limitStr;
getElement('limitValue').innerHTML = 'Limit (L): ' + (isFinite(limit) ? limit.toFixed(6) : (isNaN(limit) ? '–' : limitStr)) + '';
getElement('leftLimitValue').innerHTML = 'Left-Hand Limit (L⁻): ' + (isFinite(leftLimit) ? leftLimit.toFixed(6) : (isNaN(leftLimit) ? '–' : 'DNE')) + '';
getElement('rightLimitValue').innerHTML = 'Right-Hand Limit (L⁺): ' + (isFinite(rightLimit) ? rightLimit.toFixed(6) : (isNaN(rightLimit) ? '–' : 'DNE')) + '';
getElement('functionBehavior').innerHTML = 'Behavior near \'a\': ' + behavior + '';
// Update Summary Table
getElement('summaryFunc').textContent = functionInput;
getElement('summaryPoint').textContent = pointInput;
getElement('summaryDirection').textContent = direction === 'both' ? 'Both (+/-)' : (direction === 'positive' ? 'Right (+)' : 'Left (-)');
getElement('summaryPrecision').textContent = precision;
getElement('summaryLimit').textContent = limitStr;
getElement('resultsContainer').style.display = 'block';
// Update Chart
updateChart(functionInput, a, direction, isPointInfinity);
}
function resetCalculator() {
getElement('functionInput').value = 'x^2 + 1';
getElement('pointInput').value = '0';
getElement('directionSelect').value = 'both';
getElement('precisionInput').value = '100';
clearError('functionError');
clearError('pointError');
clearError('precisionError');
getElement('resultsContainer').style.display = 'none';
getElement('primaryResult').textContent = '–';
getElement('limitValue').innerHTML = 'Limit (L): —';
getElement('leftLimitValue').innerHTML = 'Left-Hand Limit (L⁻): —';
getElement('rightLimitValue').innerHTML = 'Right-Hand Limit (L⁺): —';
getElement('functionBehavior').innerHTML = 'Behavior near \'a\': —';
getElement('summaryFunc').textContent = '–';
getElement('summaryPoint').textContent = '–';
getElement('summaryDirection').textContent = '–';
getElement('summaryPrecision').textContent = '–';
getElement('summaryLimit').textContent = '–';
if (chartInstance) {
chartInstance.destroy();
chartInstance = null;
}
var canvas = getElement('limitChart');
var ctx = canvas.getContext('2d');
ctx.clearRect(0, 0, canvas.width, canvas.height); // Clear canvas
getElement('chartCaption').textContent = 'Function Graph Visualization';
}
function copyResults() {
var primaryResult = getElement('primaryResult').textContent;
var limitValue = getElement('limitValue').textContent.replace('Limit (L): ', ");
var leftLimitValue = getElement('leftLimitValue').textContent.replace('Left-Hand Limit (L⁻): ', ");
var rightLimitValue = getElement('rightLimitValue').textContent.replace('Right-Hand Limit (L⁺): ', ");
var behavior = getElement('functionBehavior').textContent.replace('Behavior near \'a\': ', ");
var func = getElement('summaryFunc').textContent;
var point = getElement('summaryPoint').textContent;
var direction = getElement('summaryDirection').textContent;
var precision = getElement('summaryPrecision').textContent;
var resultsText = "— Limit Calculation Results —\n\n";
resultsText += "Function: " + func + "\n";
resultsText += "Approaching Point (a): " + point + "\n";
resultsText += "Direction: " + direction + "\n";
resultsText += "Precision: " + precision + "\n\n";
resultsText += "Primary Result: " + primaryResult + "\n";
resultsText += "Limit (L): " + limitValue + "\n";
resultsText += "Left-Hand Limit (L⁻): " + leftLimitValue + "\n";
resultsText += "Right-Hand Limit (L⁺): " + rightLimitValue + "\n";
resultsText += "Behavior near 'a': " + behavior + "\n";
// Use a temporary textarea for copying
var textArea = document.createElement("textarea");
textArea.value = resultsText;
document.body.appendChild(textArea);
textArea.select();
try {
var successful = document.execCommand('copy');
var msg = successful ? 'Results copied successfully!' : 'Failed to copy results.';
// Optional: Show a temporary message to the user
// alert(msg);
} catch (err) {
// alert('Error copying results: ' + err);
}
document.body.removeChild(textArea);
}
function updateChart(functionStr, a, direction, isPointInfinity) {
var canvas = getElement('limitChart');
var ctx = canvas.getContext('2d');
ctx.clearRect(0, 0, canvas.width, canvas.height); // Clear previous drawing
var chartWidth = canvas.width;
var chartHeight = canvas.height;
var padding = 40; // Padding around the chart area
// Axis drawing parameters
var xAxisY = chartHeight / 2;
var yAxisX = chartWidth / 2;
var axisColor = '#ccc';
var labelColor = '#555';
// Determine X range for plotting
var xMinPlot, xMaxPlot, yMinPlot, yMaxPlot;
var numSamples = 200; // More samples for smooth curve
if (isPointInfinity) {
xMinPlot = a === Infinity ? 1e4 : -1e4;
xMaxPlot = a === Infinity ? 1e5 : -1e5;
if (a === -Infinity) { // Swap for negative infinity
xMinPlot = -1e5;
xMaxPlot = -1e4;
}
} else {
var range = 5; // Initial range around 'a'
if (Math.abs(a) > 100) range = Math.abs(a) * 0.1; // Adjust range for large 'a'
if (Math.abs(a) 1e-6) range = 1; // Zoom in for small 'a'
xMinPlot = a – range;
xMaxPlot = a + range;
}
var xStep = (xMaxPlot – xMinPlot) / (numSamples – 1);
var dataPoints = [];
var maxY = -Infinity;
var minY = Infinity;
for (var i = 0; i < numSamples; i++) {
var xVal = xMinPlot + i * xStep;
// Ensure we don't evaluate exactly at 'a' if it's a singularity
if (!isPointInfinity && Math.abs(xVal – a) a ? 1e-9 : -1e9); // Nudge slightly
}
// Skip points extremely close to 'a' if limit is DNE due to jump/split
if (!isPointInfinity && direction !== 'both' && Math.abs(xVal – a) 0 && dataPoints[dataPoints.length – 1].y !== null) {
dataPoints.push({ x: xVal, y: null }); // Mark gap start
}
maxY = Math.max(maxY, chartHeight); // Placeholder for max y
minY = Math.min(minY, 0); // Placeholder for min y
continue;
}
dataPoints.push({ x: xVal, y: yVal });
if (isFinite(yVal)) {
minY = Math.min(minY, yVal);
maxY = Math.max(maxY, yVal);
} else if (yVal === Infinity) {
maxY = Infinity;
} else if (yVal === -Infinity) {
minY = -Infinity;
}
}
// If no valid points, draw axes only
if (dataPoints.length === 0) {
drawAxes(ctx, chartWidth, chartHeight, padding, axisColor, labelColor);
getElement('chartCaption').textContent = 'No valid data points generated for the function.';
return;
}
// Adjust Y range to provide some padding and accommodate asymptotes
var yRange = maxY – minY;
var yPadding = yRange * 0.1; // 10% padding
// Prevent extremely small yRange from causing issues
if (yRange < 1 && !isFinite(minY) && !isFinite(maxY)) {
yPadding = 1; // Default padding if range is tiny or infinite
} else if (yRange < 1) {
yPadding = 1;
}
var plotMinY = isFinite(minY) ? minY – yPadding : -10; // Default for infinite lower bound
var plotMaxY = isFinite(maxY) ? maxY + yPadding : 10; // Default for infinite upper bound
// If only one limit value is finite, center the view around it
if(isFinite(minY) && !isFinite(maxY)) plotMaxY = minY + yRange + yPadding;
if(!isFinite(minY) && isFinite(maxY)) plotMinY = maxY – yRange – yPadding;
if (!isFinite(minY) && !isFinite(maxY)) { // Both infinite
plotMinY = -100; // Set a reasonable range
plotMaxY = 100;
}
// Scale and draw the function curve
ctx.save();
ctx.beginPath();
ctx.rect(padding, padding, chartWidth – 2 * padding, chartHeight – 2 * padding);
ctx.clip();
var firstPoint = true;
var prevY = null;
for (var i = 0; i = padding && markerX <= chartWidth – padding) {
ctx.fillStyle = '#ff7f50'; // Coral color for point 'a'
ctx.beginPath();
ctx.arc(markerX, chartHeight – padding, 5, 0, Math.PI * 2);
ctx.fill();
ctx.font = '12px Arial';
ctx.fillStyle = labelColor;
ctx.textAlign = 'center';
ctx.fillText('a', markerX, chartHeight – padding – 10);
}
}
// Update caption
getElement('chartCaption').textContent = 'Graph of ' + functionStr + ' near x=' + (isPointInfinity ? (a === Infinity ? '∞' : '-∞') : a.toFixed(2));
}
function drawAxes(ctx, width, height, padding, axisColor, labelColor) {
// Y-Axis
ctx.beginPath();
ctx.moveTo(padding, padding);
ctx.lineTo(padding, height – padding);
ctx.strokeStyle = axisColor;
ctx.lineWidth = 1;
ctx.stroke();
// X-Axis
ctx.beginPath();
ctx.moveTo(padding, height – padding);
ctx.lineTo(width – padding, height – padding);
ctx.strokeStyle = axisColor;
ctx.lineWidth = 1;
ctx.stroke();
// Origin label (if applicable and visible)
ctx.font = '12px Arial';
ctx.fillStyle = labelColor;
ctx.textAlign = 'right';
ctx.fillText('0', padding – 5, height – padding + 15);
// Labels for axes ends (approximate)
ctx.textAlign = 'center';
ctx.fillText('Y', padding – 20, padding);
ctx.fillText('X', width – padding, height – padding + 20);
}
// Initial calculation on load
document.addEventListener('DOMContentLoaded', function() {
// Use Math.js for robust function parsing and evaluation
// Ensure math.js is loaded before this script, or include it directly
// For this example, assuming math.js is available globally.
// If not, you'd need to include it via a script tag:
//
if (typeof math === 'undefined') {
console.error("Math.js library is required for function evaluation. Please include it.");
// Optionally, disable calculation or provide a fallback
getElement('functionInput').disabled = true;
getElement('pointInput').disabled = true;
getElement('directionSelect').disabled = true;
getElement('precisionInput').disabled = true;
getElement('calculateLimitButton').disabled = true;
return;
}
resetCalculator(); // Set default values and clear canvas
calculateLimit(); // Perform initial calculation
});