Limit Calculator
A limit asks: what value does this function approach as x gets closer and closer to a particular point? It is the foundation of calculus.
Last updated: August 2026 · Written and verified by Akash Pandey
Evaluate a limit
Limit
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What is a limit?
A limit describes the value a function approaches as the input approaches some value. It does not require the function to actually reach that value - only to get arbitrarily close.
lim(x→a) f(x) = L
- This means: as x gets closer to a, f(x) gets closer to L.
- The function may or may not be defined at x = a; the limit only cares about what happens near a.
When direct substitution works - and when it does not
For most polynomial and rational functions, you can simply substitute the value. If the function is defined at that point, the limit equals the function value. But direct substitution gives 0/0 for some rational functions at exactly the points that make them interesting - and 0/0 is not an answer, it is a sign the function needs simplifying first before you substitute.
Worked examples
Find the limit of 2x + 3 as x → 1
lim(x→1) (2x + 3)
- Substitute x = 1 into the function.
2(1) + 3 = 2 + 3 = 5.- The function is continuous at x = 1, so the limit is 5.
= 5
Find the limit of (x² − 4)/(x − 2) as x → 2 - where substitution fails
lim(x→2) (x² − 4)/(x − 2)
- Try direct substitution first:
(2² − 4)/(2 − 2) = 0/0, which is undefined - a signal to simplify, not a final answer. - Factor the numerator:
x² − 4 = (x − 2)(x + 2). - Cancel the shared
(x − 2)factor:(x − 2)(x + 2)/(x − 2) = x + 2, valid for every x except exactly 2. - Now substitute into the simplified form:
2 + 2 = 4. - Interpretation: the original function has a hole at x = 2 (it is undefined there), but the limit still exists and equals 4 - this is exactly what a "removable discontinuity" means.
= 4
- Try direct substitution first:
Common limit patterns
| Pattern | Limit |
|---|---|
| lim(x→0) sin(x)/x | 1 |
| lim(x→0) (1 − cos(x))/x | 0 |
| lim(x→∞) (1 + 1/x)ˣ | e ≈ 2.71828 |
| lim(x→0) (eˣ − 1)/x | 1 |
Frequently Asked Questions
When does a limit not exist?
A limit does not exist when the function approaches different values from the left and right, or when it grows without bound (approaches infinity).
What is a one-sided limit?
A one-sided limit considers the function's behaviour from only one direction - either from below (left-hand limit) or from above (right-hand limit).
What is an infinite limit?
An infinite limit means the function grows without bound as x approaches a value. We write lim f(x) = ∞ to indicate the function increases forever, though technically the limit does not exist.
What is a limit at infinity?
A limit at infinity asks what value the function approaches as x grows very large. For example, lim(x→∞) 1/x = 0, because 1/x gets closer to 0 as x increases.
When do I need to use limits?
Limits are foundational to calculus. They define derivatives, integrals, continuity, and series. They answer the question "what happens near a point" rather than "what happens at a point."
What is L'Hôpital's rule?
When a limit gives 0/0 or ∞/∞, L'Hôpital's rule says you can take the derivative of the numerator and denominator separately and try again: lim f(x)/g(x) = lim f'(x)/g'(x). Factoring and cancelling, as in the worked example above, solves many of the same 0/0 cases without needing derivatives at all.
I substituted and got 0/0 - does that mean the limit does not exist?
Not necessarily - 0/0 means direct substitution cannot answer the question, not that there is no answer. Try factoring the expression (or rationalising, for a square root) to cancel the term causing the 0/0, then substitute again into the simplified form.
Can a limit exist if the function is undefined there?
Yes. A limit only cares about what happens near the point, not at the point itself. The function can be undefined or have a different value at x = a and still have a limit - this is exactly what happens at a removable discontinuity, like a rational function with a common factor that cancels.