ex
x
and x
x
e
Limits at Infinity of
ex
x
As x approaches infinity, the rational expressions
and x take on the form
x
e
∞
∞ . Use the extended version of l’Hopital’s rule to evaluate the following limits,
if they exist.
ex
x→+∞ x
x
b) lim x
x→+∞ e
a)
lim
Solution
The extended version of l’Hopital’s rule tells us that:
f (x)
f � (x)
= lim �
x→+∞ g(x)
x→+∞ g (x)
lim
provided that limx→+∞ f (x) and limx→+∞ g(x) have the appropriate properties
�
and limx→+∞ fg� ((xx)) exists. Since we have verified that the limits in question are
of the form ∞
∞ , we may apply this rule.
a)
ex
x→+∞ x
lim
Because this is of the form ∞
∞
, we know that if the limit exists then:
ex
ex
= lim
= ∞.
x→+∞ x
x→+∞ 1
lim
As x approaches infinity,
b)
lim
x→+∞
ex
1
does as well; hence so does
ex
x .
x
ex
If the limit exists then:
lim
x→+∞
x
1
= lim x = 0.
x→+∞ e
ex
As x approaches infinity, the denominator of this rational function grows
faster than the numerator and so the limit of the quotient is 0.
We will soon see how limits such as these can be used to compare the rates
of growth of two functions.
1
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18.01SC Single Variable Calculus��
Fall 2010 ��
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