Limit Calculator
Estimate the limit of a function as x approaches a value.
Estimated numerically by approaching from both sides.
A limit is the value a function approaches as x nears a point — it exists only when the left and right sides agree. For (x² − 1) ÷ (x − 1) at x → 1, substitution gives the indeterminate 0 ÷ 0, but factoring to x + 1 shows the limit is 2, even though the function is undefined there.
Approaching a value
A limit describes what a function gets close to as its input approaches a point — even at a point where the function itself is undefined. The two-sided limit exists only when the function heads to the same value from the left and from the right. That is the whole idea behind the definition below.
the two-sided limit exists only when both one-sided limits agree
Worked example
Find the limit of (x² − 1) ÷ (x − 1) as x → 1:
- 1 Try direct substitution. At x = 1 you get (1 − 1) ÷ (1 − 1) = 0 ÷ 0 — indeterminate, so substitution alone fails.
- 2 Factor and cancel. x² − 1 = (x − 1)(x + 1), so the expression simplifies to x + 1 for x ≠ 1.
- 3 Substitute into the simplified form. x + 1 at x = 1 gives 2. Sampling 0.999 → 1.999 and 1.001 → 2.001 confirms the limit is 2.
Sampling near x = 1
Both sides of (x² − 1) ÷ (x − 1) close in on 2, even though the function is undefined at x = 1.
| x | f(x) | Side |
|---|---|---|
| 0.9 | 1.9 | left |
| 0.99 | 1.99 | left |
| 1 | undefined (0 ÷ 0) | — |
| 1.01 | 2.01 | right |
| 1.1 | 2.1 | right |
When a limit does not exist
The sides disagree. For 1 ÷ x at x = 0, the left side runs to −∞ and the right to +∞, so there is no single two-sided limit. The tool flags this disagreement.
0 ÷ 0 is indeterminate, not zero. It signals that you need algebra — factoring, rationalising, or L’Hôpital’s rule — before the limit reveals itself.
This is a numerical estimate. The tool samples just left and right of your point. That is ideal for checking work and building intuition, though a symbolic proof comes from the algebra above.