Engineering Calc Tools

Calculus

Limit Calculator

Limits at a point or infinity, one- or two-sided, with L’Hôpital’s rule shown step by step.

Tool no.
ECT-C03
Rev
1.0
Updated
Oct 2026
Method
L’Hôpital’s rule
Examples

Given

Use x as the variable. Supports + − * / ^, sqrt, exp, ln, sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, abs, pi and e.

A number, an expression such as pi/2, or inf / -inf.

Direction

Solution

Limit —
Runs in your browser. Nothing you enter is uploaded.L’Hôpital’s rule · numerical check
How to use this calculator

Type a function of x and the value x approaches. Choose a direction for one-sided limits. The calculator first tries direct substitution. If that gives 0/0 or ∞/∞ for a fraction, it applies L’Hôpital’s rule, differentiating the top and bottom. A table of values approaching the point confirms the answer and catches limits that do not exist.

Limit rules

Continuous functionslimx→a f(x) = f(a)
L’Hôpital’s rule (0/0 or ∞/∞)lim f(x)g(x) = lim f′(x)g′(x)
Two-sided limit exists only iflimx→a⁻ f = limx→a⁺ f

Worked example

lim (1 − cos x)/x² as x → 0

  1. Substituting x = 0 gives 0/0, an indeterminate form.
  2. L’Hôpital: differentiate top and bottom to get sin x / (2x). Still 0/0.
  3. Again: cos x / 2. Now substitute x = 0 to get 1/2.

Frequently asked questions

When does a limit not exist?

When the left and right limits differ (like |x|/x at 0), when the function oscillates without settling (like sin(1/x) at 0), or when it grows without bound in different directions on each side.

Is a limit of ∞ a limit that exists?

Strictly, an infinite limit means the limit does not exist as a number, but it describes how the function behaves, so it is written lim f(x) = ∞.

What about forms like 0 · ∞ or 1∞?

Rewrite them as a fraction first, for example x ln x = ln x / (1/x), so L’Hôpital’s rule applies. The calculator still gives the right value for these through its numerical check, marked as an estimate.