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l’Hôpital’s Rule Help (page 2)

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By — McGraw-Hill Professional
Updated on Apr 25, 2014

Examples

Example 1

Evaluate the limit

Indeterminate Forms 5.1 l’Hôpital’s Rule

Solution 1

This may be rewritten as

Indeterminate Forms 5.1 l’Hôpital’s Rule

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

Indeterminate Forms 5.1 l’Hôpital’s Rule

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

Indeterminate Forms 5.1 l’Hôpital’s Rule

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

Indeterminate Forms 5.1 l’Hôpital’s Rule

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

Indeterminate Forms 5.1 l’Hôpital’s Rule

Again the numerator and denominator tend to +∞ as x → +∞, so we once more apply l’Hôpital’s Rule. The limit equals

Indeterminate Forms 5.1 l’Hôpital’s Rule

We must apply l’Hôpital’s Rule two more times. We first obtain

Indeterminate Forms 5.1 l’Hôpital’s Rule

and then

Indeterminate Forms 5.1 l’Hôpital’s Rule

We conclude that

Indeterminate Forms 5.1 l’Hôpital’s Rule

You Try It : Evaluate the limit Indeterminate Forms 5.1 l’Hôpital’s Rule .

You Try It : Evaluate the limit lim x →−∞ x 4 · e x .

Example 3

Evaluate the limit

Indeterminate Forms 5.1 l’Hôpital’s Rule

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

Indeterminate Forms 5.1 l’Hôpital’s Rule

This in turn simplifies to

Indeterminate Forms 5.1 l’Hôpital’s Rule

l’Hôpital’s Rule also applies to one-sided limits. Here is an example.

Example 4

Evaluate the limit

Indeterminate Forms 5.1 l’Hôpital’s Rule

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

Indeterminate Forms 5.1 l’Hôpital’s Rule

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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