Limit Calculator
Powers ^ · multiply with * or a space · sqrt( ) ln( ) log( ) sin( ) cos( ) tan( ) e pi |x| · implicit products like 2x or x sin x work
A number, pi, e, inf or -inf
Direction
Try an example
Limit
How it was found
Direct substitution does not work here (it gives an undefined or indeterminate value such as 0/0), so the limit is estimated from f(x) at points closer and closer to the target, then extrapolated. Values that match a simple exact number (a fraction, π, e, √2) to about 7 digits are shown in exact form.
L'Hôpital's rule confirms it (0/0 form)
| x | f(x) |
|---|---|
| 0.9 | 1.9 |
| 0.99 | 1.99 |
| 0.999 | 1.999 |
| 0.9999 | 1.9999 |
| 0.99999 | 1.99999 |
| 0.999999 | 1.999999 |
| x | f(x) |
|---|---|
| 1.1 | 2.1 |
| 1.01 | 2.01 |
| 1.001 | 2.001 |
| 1.0001 | 2.0001 |
| 1.00001 | 2.00001 |
| 1.000001 | 2.000001 |
Graph
Evaluate the limit of a function as x approaches a number or ±∞, from both sides or from one. The calculator tries direct substitution, recognizes indeterminate forms such as 0/0, confirms them with L'Hôpital's rule where it can, and shows f(x) closing in from each side.
Formula
Worked Example
Substituting x = 0 gives 0/0, an indeterminate form. L'Hôpital's rule replaces the quotient with cos x ÷ 1, which equals 1 at x = 0, and the table agrees: f(0.001) = 0.99999983 from both sides.
Frequently Asked Questions
What is an indeterminate form?
A result like 0/0, ∞/∞, 0·∞, ∞ − ∞, 1^∞, 0⁰ or ∞⁰ that substitution can't settle, because the answer depends on how fast each part approaches its value. (x² − 1)/(x − 1) gives 0/0 at x = 1 but simplifies to x + 1, so its limit is 2.
When does a limit not exist?
When the left- and right-hand limits differ (a jump), when the function grows without bound in different directions (1/x at 0 goes to −∞ from the left and +∞ from the right), or when it oscillates forever, like sin(1/x) near 0.
What is the limit of (1 + 1/x)^x as x goes to infinity?
e ≈ 2.71828. It is a 1^∞ form: taking the logarithm gives x · ln(1 + 1/x), which tends to 1, so the limit is e¹.