Limit Calculator
Calculate mathematical limits, analyze function behavior at specific points, and solve calculus limit problems step by step.
Function Settings
Use x^n for powers, / for division, sin(x), cos(x), etc.
Limit Properties
- • lim[f(x) + g(x)] = lim f(x) + lim g(x)
- • lim[cf(x)] = c · lim f(x)
- • lim[f(x)g(x)] = lim f(x) · lim g(x)
- • lim[f(x)/g(x)] = lim f(x) / lim g(x) (if lim g(x) ≠ 0)
Limit Result
Limit exists
Numerical Value
Approximate value
Left Limit
x → a⁻
Right Limit
x → a⁺
Solution Steps
How it works
A limit describes the value a function approaches as its input gets arbitrarily close to some point — even if the function isn't defined exactly there. Limits are the foundation of calculus, underpinning both derivatives and integrals.
Limit
lim(x→a) f(x) = L means f(x) gets arbitrarily close to L as x → a
- a
- the value x approaches
- L
- the limit (the value f approaches)
Worked example
- lim(x→3) of (x² − 9)/(x − 3)
- Factor: (x−3)(x+3)/(x−3) = x + 3
- Substitute x = 3
The limit is 6 (even though the original is 0/0 at x = 3).
Good to know
- A limit can exist even where the function is undefined or has a hole.
- For a two-sided limit to exist, the left and right approaches must agree.
- Indeterminate forms like 0/0 often resolve by factoring or L'Hôpital's rule.
Related Calculators
Frequently Asked Questions
What is a limit in calculus?
A limit describes the value a function approaches as its input approaches some point — written lim(x→a) f(x) = L. The function doesn't need to be defined at the point itself; limits are about behavior near the point, which is what makes derivatives and continuity possible.
How do I evaluate a limit?
Try direct substitution first. If it yields a defined number, that's the limit. If you get an indeterminate form like 0/0, simplify algebraically — factor and cancel, rationalize with a conjugate, or combine fractions — and substitute again. L'Hôpital's rule is another option for 0/0 and ∞/∞.
What are one-sided limits?
A left-hand limit approaches the point from below (x→a⁻) and a right-hand limit from above (x→a⁺). The two-sided limit exists only when both agree. They differ at jump discontinuities — for example, |x|/x approaches −1 from the left and +1 from the right of zero.
What is an indeterminate form?
Forms like 0/0, ∞/∞, 0·∞, and ∞−∞ don't determine the limit by themselves — the answer depends on how fast each part grows, so more work is needed. That's exactly when factoring, rationalizing, or L'Hôpital's rule come into play.
How do limits at infinity work?
They describe end behavior: lim(x→∞) f(x) gives the horizontal asymptote if it exists. For rational functions, compare leading terms — (3x² + x)/(2x² − 5) → 3/2 as x→∞ because lower-order terms become negligible.