Limit Calculator
Evaluate limits at a point or at infinity and check them with a table of values.
Press Enter to solve · ^ for powers · sqrt() for roots · separate a system with a comma
The limit calculator finds the value a function approaches as x gets close to a number or to infinity. Type x -> 0 sin(x)/x or x -> Infinity (3x^2+1)/x^2 and the solver tries direct substitution, handles indeterminate forms and shows a table of nearby values.
How to use the limit calculator
- Write the variable and the value it approaches, then the expression:
x -> 2 (x^2-4)/(x-2). - Use
Infinityor-Infinityfor limits at infinity. - Press Enter.
How limits are evaluated
- Direct substitution. If plugging in the value gives a real number, that is the limit.
- Indeterminate form. If you get 0/0 or ∞/∞, simplify (factor and cancel) or use L’Hôpital’s rule.
- Limits at infinity. Compare the highest powers in the numerator and denominator.
Example: lim x→2 (x² − 4)/(x − 2)
- Substitution gives 0/0, an indeterminate form.
- Factor: (x − 2)(x + 2)/(x − 2) = x + 2.
- Substitute again: 2 + 2 = 4.
Questions about limits
What is lim x→0 of sin(x)/x?
It equals 1. This is one of the most important limits in calculus and is used to prove the derivative of sin x.
What does it mean if a limit does not exist?
The function approaches different values from the left and right, grows without bound, or oscillates. The value table makes this easy to see.
When can I use L’Hôpital’s rule?
Only for the indeterminate forms 0/0 or ∞/∞. Then the limit of f/g equals the limit of f′/g′.