Limit Calculator with Steps
Type a function and the value x approaches. Calvo shows every step: direct substitution, factoring and cancelling, L’Hôpital’s rule, limits at infinity and one-sided limits, then checks the answer with numbers.
Find a limit step by step
Use * for multiplication (2x also works), ^ for powers, sqrt(), ln(), e^x, sin(), cos(), tan(), abs(). For infinity type inf or -inf. One letter only.
Methods Calvo uses
| Situation | Method |
|---|---|
| Function is defined at the point | Direct substitution |
| 0/0 with polynomials | Factor, cancel, substitute |
| 0/0 or ∞/∞ with other functions | L’Hôpital’s rule (differentiate top and bottom) |
| x → ∞ with a fraction of polynomials | Compare the highest powers |
| a/0 (blows up) | Check the sign from each side |
| 0 × ∞ | Rewrite as a fraction, then L’Hôpital |
Worked example: 0/0 form
Find the limit of (x² − 4)/(x − 2) as x → 2. Substituting gives 0/0, so factor: (x − 2)(x + 2)/(x − 2). Cancel (x − 2) to get x + 2. Now substitute x = 2: the limit is 4.
How Calvo checks the answer
After finding an answer with exact rules, Calvo puts in numbers very close to the point (from both sides, or very large values for infinity) and compares. If a limit cannot be found exactly it says so and shows a clearly marked numerical estimate instead of guessing.
What is not supported
One variable only. Piecewise functions, limits of sequences defined by sums, and expressions with several letters are not supported. Trigonometric limits at special points are found by L’Hôpital or numerically.
FAQ
How do I find a limit step by step?
First substitute the value. If you get a number, that is the limit. If you get 0/0 or infinity/infinity, factor and cancel, use L'Hopital's rule, or compare the highest powers for limits at infinity. Calvo shows each step.
What is L'Hopital's rule?
If a limit gives 0/0 or infinity/infinity, the limit of f/g equals the limit of f'/g' (differentiate the top and bottom separately), provided that limit exists.
What is the limit of sin(x)/x as x approaches 0?
It is 1. Substitution gives 0/0, and L'Hopital's rule gives cos(x)/1, which equals 1 at x = 0.
When does a limit not exist?
When the left-hand and right-hand limits are different, or when the function oscillates without settling. For example, 1/x as x approaches 0 has a left limit of minus infinity and a right limit of plus infinity.
How do I find a limit at infinity?
For a fraction of polynomials compare degrees: smaller top degree gives 0, equal degrees give the ratio of leading coefficients, and larger top degree gives infinity. Other functions use L'Hopital's rule or a numerical check.