Limit Calculator
How to use the limit calculator
Enter a function f(x) and the value a that x approaches. Choose a two-sided limit or a one-sided limit from the left or right. The calculator evaluates the limit and shows the steps, including the left- and right-hand limits for two-sided limits.
Worked example
lim (x→0) sin(x)/x = 1. Although sin(x)/x is undefined at x = 0, values of x near 0 give ratios close to 1, and the squeeze theorem confirms the limit is 1.
When are limits used?
Limits define derivatives and integrals, describe asymptotes and continuity, and evaluate indeterminate forms such as 0/0.
For an indeterminate form worked in full, see the limit of (eˣ−1−x)/x² as x→0 solved with series and L’Hôpital.
Frequently asked questions
A two-sided limit exists only when the left-hand and right-hand limits are equal. If they disagree, the two-sided limit does not exist.
Yes. Enter inf or -inf for a to find the limit as x approaches positive or negative infinity.
A limit can still exist when f(a) is undefined. The limit depends on values near a, not on the value at a.
How a limit is actually found
Start with direct substitution. If putting x = a gives a plain number, that number is the limit. If it gives 0/0, a common factor is usually hiding the real behaviour, so factor, cancel the shared factor, and substitute again. For square roots, multiplying top and bottom by the conjugate is the standard trick.
This is L’Hopital’s rule: replace a hard limit with the limit of the derivatives. It applies only to those two forms, so confirm the substitution gives one of them first.
Example: Find lim (x -> 2) (x^2 – 5x + 6)/(x – 2)
Step 1, Try substitution: Putting x = 2 gives (4 – 10 + 6)/0 = 0/0, an indeterminate form, so keep working.
Step 2, Factor the top: x^2 – 5x + 6 = (x – 2)(x – 3).
Step 3, Cancel the shared factor: For x near 2 but not equal to 2, (x – 2) is not zero, so it cancels and leaves x – 3.
Step 4, Substitute again: The limit is 2 – 3 = -1. Values confirm it: at x = 1.99 the fraction is about -1.01, and at x = 2.01 about -0.99.
When limits are the right tool
Reach for a limit whenever a formula breaks down at the exact point you care about but behaves well nearby: a hole in a graph, or the long run behaviour of a function as x grows.
Common mistakes
- Reading 0/0 as an answer: 0/0 means indeterminate, not 0, not 1 and not no limit. Simplify first.
- Cancelling, then claiming the value exists: Cancelling finds the limit, but the function can still be undefined at a itself.
- Ignoring the two sides: A jump or asymptote can make the left and right limits disagree, so no two-sided limit exists.
- Using L’Hopital too early: Applied to a form that is not 0/0 or infinity over infinity, it gives a wrong answer.
More questions
It means substitution alone cannot decide the answer. Forms like 0/0 can hide any limit at all, so you must simplify or factor to reveal it.
From the right, 1/x grows toward positive infinity. From the left, it falls toward negative infinity. The two sides disagree completely, so there is no single two-sided limit at 0.
Evaluate at values closing in on a from both sides, such as a – 0.01 and a + 0.01. If both sides settle toward the same number, that is very likely the limit.