Limit Calculator

Study what a function approaches from the left, right, or at infinity, even when direct substitution fails.

Drag & drop or upload an image or PDF

Enter a limit, such as limit (x^2 - 4)/(x - 2) as x approaches 2

Try an example

A limit describes nearby behavior

A limit asks what a function approaches, not necessarily what the function equals at the target point. That is why a removable hole can have a limit even though the original expression is undefined there.

Direct substitution is the first check, not always the last step. An indeterminate form may call for factoring, rationalization, a standard limit, dominant-term analysis, or L'Hopital's rule when its hypotheses are satisfied.

How to use the limit calculator

Enter the problem as written

Type or paste the full expression. You can also upload a clear photo or PDF and check the extracted text before solving.

Read the working, not only the answer

Each transformation is separated and explained so you can compare it with your own method.

Ask about any step

Continue in the same solution to request another method, check a restriction, or ask why a rule applies.

Agreement of one-sided limits

A two-sided limit exists only when the values approached from the left and right agree.

x \to a lim f (x) = L ⟺ x \to a^{-} lim f (x) = x \to a^{+} lim f (x) = L

A reliable way to work through it

Try direct substitution

If it produces a defined value and the function is continuous there, the limit is immediate.

Resolve the indeterminate form

Factor, rationalize, use a standard limit, or apply a justified theorem instead of treating 0/0 as an answer.

Check direction and domain

Compare left and right behavior for jumps, roots, logarithms, absolute values, and piecewise functions.

Worked example

Find lim as x approaches 2 of (x² - 4)/(x - 2)

\frac{2 ^{2} - 4}{2 - 2} = \frac{0}{0}

Direct substitution gives an indeterminate form.

x^{2} - 4 = (x - 2) (x + 2)

Factor the difference of squares.

\frac{( x - 2 ) ( x + 2 )}{x - 2} = x + 2 (x \neq = 2)

Cancel only for nearby values where x is not 2.

lim_{x \to 2} (x + 2) = 4

Now substitute into the simplified expression.

The limit is 4 even though the original expression is undefined at x = 2.

Common mistakes to check

Stopping at 0/0

The form 0/0 signals that more work is needed. It is not the value of the limit.

Ignoring one-sided behavior

A two-sided limit fails when the left-hand and right-hand limits disagree.

Using L'Hopital's rule automatically

The rule applies only to justified 0/0 or infinity/infinity forms under the required conditions.

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Questions students ask

Can a limit exist when the function is undefined?

Yes. A limit describes nearby behavior, so the value at the target can be missing or different.

When does a two-sided limit exist?

The left-hand and right-hand limits must approach the same value.

What does DNE mean?

It means the limit does not exist as one value, often because of a jump, oscillation, or incompatible one-sided behavior.

When can I use L'Hopital's rule?

Use it only after confirming a 0/0 or infinity/infinity indeterminate form and the needed differentiability conditions.