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Objectives

  • Recognize when to apply L’Hôpital’s rule.
  • Identify indeterminate forms produced by quotients, products, subtractions, and powers, and apply L’Hôpital’s rule in each case.
  • Describe the relative growth rates of functions.

Summary

Derivatives can be used to help us evaluate indeterminate limits of the form 0 0 through L'Hôpital's Rule, by replacing the functions in the numerator and denominator with their tangent line approximations. In particular, if f ( a ) = g ( a ) = 0 and f and g are differentiable at a , L'Hôpital's Rule tells us that

lim x → a f ( x ) g ( x ) = lim x → a f ′ ( x ) g ′ ( x ) .

When we write x → ∞ , this means that x is increasing without bound. Thus, lim x → ∞ f ( x ) = L means that we can make f ( x ) as close to L as we like by choosing x to be sufficiently large. Similarly, lim x → a f ( x ) = ∞ , means that we can make f ( x ) as large as we like by choosing x sufficiently close to a .

A version of L'Hôpital's Rule also helps us evaluate indeterminate limits of the form ∞ ∞ . If f and g are differentiable and both approach zero or both approach ± ∞ as x → a (where a is allowed to be ∞ ), then

lim x → a f ( x ) g ( x ) = lim x → a f ′ ( x ) g ′ ( x ) .