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L'Hôpital's Rule

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Tap a term to see what it means, or .

Worked examples

0/0, applying the rule twice

  1. Plugging in x = 0 gives 0/0, an indeterminate form:
  2. Differentiate the numerator and the denominator separately:
  3. It is still 0/0 at x = 0. Apply the rule again:
  4. Now plugging in works:

∞/∞

  1. As x grows, the numerator and the denominator both go to infinity: ∞/∞.
  2. Differentiate each part: the derivative of ln x is 1/x, and that of x is 1.
  3. As x goes to infinity, 1/x goes to zero:

Common mistake: using the rule without an indeterminate form

  1. Here plugging in directly already works, because there is no indeterminate form:
  2. Applying the rule anyway would give a wrong result:

The rule only holds for 0/0 or ∞/∞. Before differentiating, plug in the point and check that the form really is indeterminate.

What each term means

Indeterminate Form
The rule applies when the limit of a fraction f(x)/g(x) gives an indeterminate form, such as 0/0 or ±∞/±∞.
Limit
The limit of the new fraction is taken as 'x' approaches the same point 'c'.
Derivative of the Numerator
Take the derivative of the original numerator, f(x), which gives f'(x).
Derivative of the Denominator
Take the derivative of the original denominator, g(x), which gives g'(x). Note that this is not the derivative of the quotient.