L'Hospital's Rule Calculator

Use L'Hospital's Rule (Bernoulli's Rule) to evaluate limits that produce indeterminate forms such as 0/0 or ∞/∞.

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Step-by-Step Solution

What is L'Hospital's Rule?

L'Hospital's Rule is a technique used to evaluate limits that initially result in an indeterminate form. The rule states that if:

lim f(x)/g(x)

approaches either:

  • 0 / 0
  • ∞ / ∞

then the limit can often be evaluated by differentiating the numerator and denominator separately:

lim f'(x) / g'(x)

provided the resulting limit exists.

Common Example

Evaluate:

limx→0 sin(x)/x

Since:

sin(0)=0 and x=0

the expression becomes 0/0. Applying L'Hospital's Rule:

d/dx[sin(x)] = cos(x)
d/dx[x] = 1

Therefore:

limx→0 cos(x)/1 = 1

When Can L'Hospital's Rule Be Used?

L'Hospital's Rule applies only when direct substitution produces an indeterminate form. The most common indeterminate forms are:

  • 0 / 0
  • ∞ / ∞

If the limit does not produce one of these forms, L'Hospital's Rule should not be applied directly. In some cases algebraic simplification must be performed first.

Formula

If

limx→a f(x)/g(x)

results in 0/0 or ∞/∞, then

limx→a f(x)/g(x) = limx→a f′(x)/g′(x)

provided the derivatives exist and the resulting limit exists.

Solved Example 1

Evaluate:

limx→0 sin(x)/x

Substitute x = 0:

sin(0) / 0 = 0 / 0

Apply L'Hospital's Rule:

Derivative of numerator:

d/dx[sin(x)] = cos(x)

Derivative of denominator:

d/dx[x] = 1

Therefore:

limx→0 cos(x) = 1

Answer = 1

Solved Example 2

Evaluate:

limx→0 (ex − 1)/x

Substituting x = 0 gives:

(1 − 1)/0 = 0/0

Apply L'Hospital's Rule:

d/dx[ex − 1] = ex

d/dx[x] = 1

Now evaluate:

limx→0 ex = 1

Answer = 1

Benefits of L'Hospital's Rule

  • Simplifies difficult limit problems.
  • Works for many indeterminate forms.
  • Reduces algebraic manipulation.
  • Useful in calculus, engineering, and physics.
  • Provides a systematic way to evaluate limits.

Frequently Asked Questions

What is L'Hospital's Rule?

L'Hospital's Rule is a calculus technique used to evaluate limits that result in indeterminate forms such as 0/0 or ∞/∞.

What is Bernoulli's Rule?

Bernoulli's Rule is another name commonly used for L'Hospital's Rule in some mathematical literature.

Can L'Hospital's Rule be applied repeatedly?

Yes. If the first differentiation still produces an indeterminate form, the rule may be applied again.

Does L'Hospital's Rule work for all limits?

No. It applies only to appropriate indeterminate forms and when the required derivatives exist.

Why do we differentiate both numerator and denominator?

Differentiation often removes the indeterminate form and transforms the limit into one that is easier to evaluate.