CalcViz
All tools

Differentiate

Derivative calculatorSecond derivativePartial derivativesImplicit differentiationChain ruleTangent lineLinear approximationCritical pointsInflection pointsMean value theorem

Integrate

Integral calculatorRiemann sumsArea under a curveAverage value

Limits & series

Limit calculatorTaylor series

Graph

Calculus grapher

Learn

What is a derivative?What is an integral?Derivative rulesIntegration rulesThe power rule

Limit Calculator

Find limits step by step: substitution first, then L’Hôpital’s rule or leading terms, and always a two-sided table of values so you can see the function close in.

Reads as sin⁡(x)x\frac{\sin\left(x\right)}{x}
Try:
-4-3-2-11234-0.20.20.40.60.81

How limits are evaluated

  1. Try direct substitution. If ff is continuous at aa, the limit is just f(a)f(a). Polynomials, exe^x, sine and cosine are continuous everywhere.
  2. If you get 00\tfrac00 or ∞∞\tfrac{\infty}{\infty}, the form is indeterminate. Factor and cancel (x2−9x−3=x+3\tfrac{x^2-9}{x-3} = x+3), or differentiate top and bottom with L’Hôpital’s rule.
  3. At infinity, rational functions are governed by their leading terms: compare the degrees of top and bottom.
  4. For 1∞1^\infty and 000^0 forms, take logarithms first; that is how lim⁡x→∞(1+1x)x=e\lim_{x\to\infty}(1+\tfrac1x)^x = e is shown.
  5. Check both sides. The two-sided limit exists only if the left and right limits agree.

Why limits matter

Every idea in calculus is a limit in disguise. The derivative is the limit of secant slopes (drag it on what is a derivative?), and the definite integral is the limit of Riemann sums as the rectangles get thinner.

Questions students ask

How do I enter infinity?

Type inf, infinity or ∞ in the "approaches" box, or -inf for negative infinity.

What does “0/0 indeterminate” mean?

Plugging in gives 00\frac{0}{0}, which could be any value. It is a signal to do more work: factor and cancel, rationalise, or use L’Hôpital’s rule. The calculator shows which one it used.

When does a limit not exist?

When the left-hand and right-hand limits differ (like ∣x∣x\frac{|x|}{x} at 0), when the function oscillates forever (like sin⁡1x\sin\frac1x at 0), or when it blows up in opposite directions (like 1x\frac1x at 0). The two-sided table makes this visible.

What is L’Hôpital’s rule?

If f(x)g(x)\frac{f(x)}{g(x)} gives 00\frac00 or ∞∞\frac{\infty}{\infty}, the limit equals lim⁡f′(x)g′(x)\lim \frac{f'(x)}{g'(x)} (if that exists). The calculator applies it repeatedly when needed.

Why show a table of values?

A limit is about what the function approaches, not its value at the point. The table evaluates the function closer and closer from both sides so you can see the trend yourself, which is how limits are introduced in most textbooks.