Limit Calculator

Finite, one-sided and infinite limits, including 0/0 and ∞/∞ forms, estimated numerically and confirmed symbolically when possible.

Your result will appear here

Fill in the fields and press Calculate, or pick one of the examples.

About the Limit Calculator

Estimates limits at a point, from one side, or at ±infinity. The function is evaluated ever closer to the point and the trend is extrapolated (Richardson extrapolation), which handles removable singularities and indeterminate forms such as 0/0 and ∞/∞.

When the symbolic engine can also compute the limit and agrees with the numerical estimate, the exact value is shown and the result is marked Exact. Otherwise the result is marked Approximate.

Limitations

A numerical estimate is evidence, not a proof. Functions that oscillate infinitely often or converge extremely slowly can be reported as “does not exist”.
One variable only.

How to use it

Type the function.
Enter the point the variable approaches (a number, an expression, inf or -inf).
Choose two-sided or one-sided and press Calculate.

Example

lim x→0 sin(x)/x

→ 1

Formulas used

\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L

A two-sided limit exists only when both one-sided limits agree.

\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}

L’Hôpital’s rule for 0/0 and ∞/∞ forms (when the right-hand limit exists).

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About the Limit Calculator

Limitations

How to use it

Formulas used

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About the Limit Calculator

Limitations

How to use it

Formulas used