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Indeterminate Forms
CHAPTER 5
5.2 Other Indeterminate Forms

5.2.1 INTRODUCTION

By using some algebraic manipulations, we can reduce a variety of indeterminate
limits to expressions which can be treated by l’Hôpital’s Rule. We explore some of
these techniques in this section.
5.2.2 WRITING A PRODUCT AS A QUOTIENT

The technique of the first example is a simple one, but it is used frequently.
EXAMPLE 5.7

Evaluate the limit 2 3x
lim x · e .
x→−∞

SOLUTION 2 3x → 0. So the limit is indeterminate of the
Notice that x →+∞ while e
form 0 ·∞. We rewrite the limit as
2
x
TEAMFLY .
lim
x→−∞ e −3x
Now both numerator and denominator tend to infinity and we may apply
l’Hôpital’s Rule. The result is that the limit equals
2x
lim
x→−∞ −3e −3x .

Again the numerator and denominator both tend to infinity so we apply
l’Hôpital’s Rule to obtain:
lim 2 .
x→−∞ 9e −3x
It is clear that the limit of this last expression is zero. We conclude that
3x
lim x · e
x→−∞ = 0.
√
You Try It: Evaluate the limit lim x→+∞ e − x · x.

5.2.3 THE USE OF THE LOGARITHM
The natural logarithm can be used to reduce an expression involving exponentials
to one involving a product or a quotient.

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