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  • 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.

In this section, we examine a powerful tool for evaluating limits. This tool, known as L’Hôpital’s rule    , uses derivatives to calculate limits. With this rule, we will be able to evaluate many limits we have not yet been able to determine. Instead of relying on numerical evidence to conjecture that a limit exists, we will be able to show definitively that a limit exists and to determine its exact value.

Applying l’hôpital’s rule

L’Hôpital’s rule can be used to evaluate limits involving the quotient of two functions. Consider

lim x a f ( x ) g ( x ) .

If lim x a f ( x ) = L 1 and lim x a g ( x ) = L 2 0 , then

lim x a f ( x ) g ( x ) = L 1 L 2 .

However, what happens if lim x a f ( x ) = 0 and lim x a g ( x ) = 0 ? We call this one of the indeterminate forms    , of type 0 0 . This is considered an indeterminate form because we cannot determine the exact behavior of f ( x ) g ( x ) as x a without further analysis. We have seen examples of this earlier in the text. For example, consider

lim x 2 x 2 4 x 2 and lim x 0 sin x x .

For the first of these examples, we can evaluate the limit by factoring the numerator and writing

lim x 2 x 2 4 x 2 = lim x 2 ( x + 2 ) ( x 2 ) x 2 = lim x 2 ( x + 2 ) = 2 + 2 = 4 .

For lim x 0 sin x x we were able to show, using a geometric argument, that

lim x 0 sin x x = 1 .

Here we use a different technique for evaluating limits such as these. Not only does this technique provide an easier way to evaluate these limits, but also, and more important, it provides us with a way to evaluate many other limits that we could not calculate previously.

The idea behind L’Hôpital’s rule can be explained using local linear approximations. Consider two differentiable functions f and g such that lim x a f ( x ) = 0 = lim x a g ( x ) and such that g ( a ) 0 For x near a , we can write

f ( x ) f ( a ) + f ( a ) ( x a )

and

g ( x ) g ( a ) + g ( a ) ( x a ) .

Therefore,

f ( x ) g ( x ) f ( a ) + f ( a ) ( x a ) g ( a ) + g ( a ) ( x a ) .
Two functions y = f(x) and y = g(x) are drawn such that they cross at a point above x = a. The linear approximations of these two functions y = f(a) + f’(a)(x – a) and y = g(a) + g’(a)(x – a) are also drawn.
If lim x a f ( x ) = lim x a g ( x ) , then the ratio f ( x ) / g ( x ) is approximately equal to the ratio of their linear approximations near a .

Since f is differentiable at a , then f is continuous at a , and therefore f ( a ) = lim x a f ( x ) = 0 . Similarly, g ( a ) = lim x a g ( x ) = 0 . If we also assume that f and g are continuous at x = a , then f ( a ) = lim x a f ( x ) and g ( a ) = lim x a g ( x ) . Using these ideas, we conclude that

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

Note that the assumption that f and g are continuous at a and g ( a ) 0 can be loosened. We state L’Hôpital’s rule formally for the indeterminate form 0 0 . Also note that the notation 0 0 does not mean we are actually dividing zero by zero. Rather, we are using the notation 0 0 to represent a quotient of limits, each of which is zero.

L’hôpital’s rule (0/0 case)

Suppose f and g are differentiable functions over an open interval containing a , except possibly at a . If lim x a f ( x ) = 0 and lim x a g ( x ) = 0 , then

lim x a f ( x ) g ( x ) = lim x a f ( x ) g ( x ) ,

assuming the limit on the right exists or is or . This result also holds if we are considering one-sided limits, or if a = and .

Practice Key Terms 2

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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