Examples
Example 1
Evaluate the limit
Solution 1
This may be rewritten as
Notice that the numerator tends to −∞ and the denominator tends to ±∞ as x → 0. Thus the quotient is indeterminate at 0 of the form −∞/+ ∞. So we may apply l’Hôpital’s Rule for infinite limits to see that the limit equals
Yet another version of l’Hôpital’s Rule, this time for unbounded intervals, is this: Let f and g be differentiable functions on an interval of the form [ A , +∞). If lim _{x →+∞} f ( x ) = lim _{x →+∞} g ( x ) = 0 or if lim _{x →+∞} f ( x ) = ±∞ and lim _{x →+∞} g ( x ) = ±∞, then
provided that this last limit exists either as a finite or infinite limit. The same result holds for f and g defined on an interval of the form (−∞, B ] and for the limit as x → −∞.
Example 2
Evaluate
Solution 2
We first notice that both the numerator and the denominator tend to +∞ as x → +∞. Thus the quotient is indeterminate at +∞ of the form +∞/+ ∞. Therefore the new version of l’Hôpital’s Rule applies and our limit equals
Again the numerator and denominator tend to +∞ as x → +∞, so we once more apply l’Hôpital’s Rule. The limit equals
We must apply l’Hôpital’s Rule two more times. We first obtain
and then
We conclude that
You Try It : Evaluate the limit .
You Try It : Evaluate the limit lim _{x →−∞} x ^{4} · e ^{x} .
Example 3
Evaluate the limit
Solution 3
We note that both numerator and denominator tend to 0, so the quotient is indeterminate at −∞ of the form 0/0. We may therefore apply l’Hôpital’s Rule. Our limit equals
This in turn simplifies to
l’Hôpital’s Rule also applies to onesided limits. Here is an example.
Example 4
Evaluate the limit
Solution 4
Both numerator and denominator tend to zero so the quotient is indeterminate at 0 of the form 0/0. We may apply l’Hôpital’s Rule; differentiating numerator and denominator, we find that the limit equals
You Try It : How can we apply l’Hôpital’s Rule to evaluate lim _{x →0} + x · ln x ?
Find practice problems and solutions for these concepts at: Indeterminate Forms Practice Test.
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