Limit Calculator

Enter a function of x and the value x approaches - see the limit estimated numerically from both sides.

Limit Calculator

Live

f(x) =

x approaches

Estimated limit

From the left

3.9999

From the right

4.0001

x	f(x)

Estimates the limit numerically by evaluating f(x) at points very close to the target - not a symbolic solver.

How it works

Evaluate f(x) as x gets closer and closer to the target from both sides If both sides converge to the same value, that value is the limit

Example

lim(x→2) of (x²-4)/(x-2): direct substitution gives 0/0, but evaluating just above and below x=2 shows both sides converging to 4.

Step-by-step guide

Enter your function in terms of x.
Enter the value x approaches.
Read the estimated limit, and check the table to see the values converging.

Why this is numerical, not symbolic

This calculator estimates a limit the same way you'd approach it by hand with a table of values - plugging in numbers closer and closer to the target and watching where the output heads. It's a genuinely reliable method for most functions you'll encounter in an introductory calculus course, but it's fundamentally different from a symbolic solver that proves a limit algebraically (like factoring out a common term or applying L'Hopital's rule) - for functions with subtle behavior, a numerical estimate can occasionally be misleading, which is exactly why the table of approaching values is shown rather than just a single final number.

Common mistakes

Assuming a limit doesn't exist just because direct substitution gives an undefined result (like 0/0) - this is exactly the situation limits are built to handle, and the function often still approaches a well-defined value.

Not checking that both sides agree - if the left-side and right-side values are heading toward different numbers, the two-sided limit doesn't exist even if each one-sided limit does.

Frequently asked questions

What does it mean if the left and right sides give different values?

It means the two-sided limit doesn't exist at that point, even though one-sided limits from each direction might still exist individually - this commonly happens at a jump discontinuity in the function.

Can this calculator handle limits at infinity?

Not directly through the "x approaches" field, but you can approximate it by entering a very large number (like 100000) to see the general trend as x grows large.

Why use a table instead of just showing one number?

Seeing the sequence of values as they approach the target builds real intuition for what a limit actually means, and makes it obvious whether the values are genuinely converging or behaving erratically.

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