Limit Calculator
- Write the limit: lim x → ∞ of x².
- Look at what happens as x grows without bound. Direct substitution is not possible, so the function is studied for very large values.
- The limit is ∞: the function increases without bound.
- Limits at infinity: check what happens to the function as x grows without bound.
- Numerical check: sample values close to the point, or very large values, confirm the trend.
- Large values: f(1000) = 1000000, f(10000) = 100000000, f(10000000) = 1×10^14.
- The values keep growing in size, so there is no finite limit.
Limit Calculator: Find the Limit of a Function with Steps
This free limit calculator finds what a function approaches as its variable gets close to a number, or as it grows without bound. It works with fractions, roots, powers, trigonometric, exponential and logarithmic functions, gives the exact answer where there is one, and shows the left-hand and right-hand limits side by side. A graph marks the limit so you can see what the number means.
Quick answer: the limit of sin(x)/x as x approaches 0 is 1, even though the function is not defined at 0. The limit of x² as x approaches infinity is ∞.
What Is a Limit?
A limit tells you where a function is heading. You pick a point, look at the values of the function on both sides of it, and see which number they close in on. The function does not need to be defined at the point, and when it is, the value there may be different. Limits are the foundation of calculus: the derivative and the integral are both defined with them.
Left-Hand and Right-Hand Limits
Approaching a point from the left means using values slightly below it, and from the right means using values slightly above it. When the two agree, that shared number is the limit. When they differ, as with 1/x at 0, the limit does not exist, but each side may still have a value of its own. Choose Both sides, From the left or From the right in the input panel to look at each case.
Indeterminate Forms and L'Hôpital's Rule
Substituting the point sometimes gives 0/0 or ∞/∞. These results do not tell you the limit, and the function needs more work. You can factor and cancel, multiply by a conjugate, or use L'Hôpital's rule, which replaces the fraction with the derivative of the top over the derivative of the bottom. Products such as 0 · ∞ are first rewritten as fractions, and powers with a variable exponent are handled with the natural log. The Rules applied box lists what your limit needed. Some common limits are shown below.
| Limit | Value | Why |
|---|---|---|
| lim x→2 of (x² − 4)/(x − 2) | 4 | Factor and cancel |
| lim x→0 of sin(x)/x | 1 | Standard result |
| lim x→0 of (1 − cos x)/x² | 1/2 | L'Hôpital's rule twice |
| lim x→∞ of (2x² + 1)/(x² − 3) | 2 | Ratio of leading terms |
| lim x→∞ of (1 + 1/x)ˣ | e | Power with a variable exponent |
| lim x→0⁺ of 1/x | ∞ | Grows without bound |
| lim x→0 of 1/x | Does not exist | Sides differ |
| lim x→0⁺ of x ln(x) | 0 | Product rewritten as a fraction |
Limits at Infinity
To see what happens for very large values, type inf in the approaches box, or -inf for very large negative values. When a function levels off at a number, its graph has a horizontal asymptote at that height. For a fraction made of polynomials, compare the highest powers: equal powers give the ratio of the leading coefficients, a larger power on the bottom gives 0, and a larger power on the top gives an infinite limit.
Continuity
A function is continuous at a point when the limit exists, the function is defined there, and the two are equal. The calculator shows f of the point next to the limit so the comparison is easy. A hole in the graph, drawn as an open circle, means the limit exists but the function has no value there. A jump or a vertical asymptote means the function is not continuous at that point.
How to Enter a Function and a Point
Type the function the way you would write it on paper. Use ^ for powers, so x squared is x^2, and write square roots as sqrt(x) and the exponential function as e^x. Functions such as sin, cos, tan and ln need brackets around their input. The point can be a number or an expression like pi/2 or sqrt(2). For related work, the polynomial calculator, the scientific calculator and the math solver can help with the algebra around it.
How to Use This Limit Calculator
Type your function
Enter it in the box, or use the shortcut buttons for powers, roots and functions.
Choose the point
Type a number, use the slider, or press ∞ to see what happens for very large values.
Pick a direction
Use both sides, or check the left and right limits one at a time.
Read the results
See the exact answer, the one-sided limits, the working and the graph.