Limit Calculator

A limit describes the value a function approaches as x gets closer to a point, even if the function is not defined there. This limit calculator tries direct substitution first, uses L'Hôpital's rule for 0/0 forms, compares degrees for limits at infinity, and checks the answer by approaching from both sides.

Type inf for infinity.
Try:

How to use the limit calculator

  1. Type the function, such as sin(x)/x.
  2. Enter the value x approaches, or type inf for infinity.
  3. Press Calculate to see the limit, the method and the graph.

Formula

lim⁡x→af(x)g(x)=lim⁡x→af′(x)g′(x) when 00\lim_{x\to a} \frac{f(x)}{g(x)} = \lim_{x\to a} \frac{f'(x)}{g'(x)} \text{ when } \frac{0}{0}

Worked example: sin(x)/x as x → 0

  • f(x) =: sin(x)/x
  • x approaches: 0

✓ Answer checked

lim⁡x→0sin⁡(x)x=1\lim_{x \to 0} \frac{\sin\left(x\right)}{x} = 1
−4−3−2−11234−0.20.20.40.60.81limit 1f(x)
As x approaches 0 from both sides, f(x) approaches 1.
Step-by-step working (4 steps)

  1. Read the function

    This is how the input was understood.

    f(x)=sin⁡(x)xf(x) = \frac{\sin\left(x\right)}{x}

  2. 0/0 form L'Hôpital's rule

    Substituting x = 0 gives 0/0, which is indeterminate. Apply L'Hôpital's rule: differentiate the top and the bottom separately.

    sin⁡(x)x→00\frac{\sin\left(x\right)}{x} \to \frac{0}{0}

  3. Differentiate top and bottom

    Take the derivative of each part.

    cos⁡(x)1\frac{\cos\left(x\right)}{1}

  4. Substitute

    Now substitute x = 0.

    11=1\frac{1}{1} = 1
Formulas used
lim⁡fg=lim⁡f′g′ when 00\lim \frac{f}{g} = \lim \frac{f'}{g'} \text{ when } \frac{0}{0}
How this was checked
  • ✓ Numerically, f(0 − 0.0001) ≈ 1 and f(0 + 0.0001) ≈ 1, both close to 1.

Frequently asked questions

What is the limit of sin(x)/x as x approaches 0?

It is 1. Substituting gives 0/0, and L'Hôpital's rule gives cos(0)/1 = 1.

When does a limit not exist?

When the values from the left and from the right approach different numbers, or grow without bound in different directions, like 1/x at 0.

What is L'Hôpital's rule?

If substituting gives 0/0, the limit of f/g equals the limit of f′/g′, the ratio of the derivatives.

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