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CALCULUS 1 PROBLEMS & SOLUTIONS

5.4.2
More Indeterminate Forms


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Review


1. The Indeterminate Quotient Forms

In Section 1.1.3 we discussed the indeterminate quotient form 0/0, and in Section 1.1.5.3 we handled
the indeterminate quotient form ¥/¥. We gather them together here:



Evaluation

We can use L'Hôpital's Rule in evaluating limits of such forms.

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2. The Indeterminate Sum And Difference Forms

In Section 1.1.5.2 we encountered the indeterminate difference form ¥ – ¥. The indeterminate sum
form is – ¥ + ¥, which is just the difference form written in a different order. We put them together here:

– ¥ + ¥, ¥ – ¥.

Evaluation

For example:



To evaluate a limit of any one of such forms, we change them, by algebraic means, to an
indeterminate quotient form, for which we can use L'Hôpital's Rule.

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3. The Indeterminate Product Forms

Consider these three trivial limits:



All three are in the form 0 . ¥. For the same reason as with the indeterminate quotient forms (see
Sections 1.1.3 and 1.1.5.3), 0 . ¥ is an indeterminate form. The indeterminate product forms are:

0 . ¥, 0 . (– ¥), ¥ . 0, (– ¥) . 0.

Evaluation

For example:



To evaluate a limit of any one of such forms, we convert it into an indeterminate quotient form, for
which we can use L'Hôpital's Rule, as follows. Suppose lim_x®a f(x) = 0 and lim_x®a g(x) = ¥. So
lim_x®a ( f(x)g(x)) = lim_x®a ( f(x)/(1/g(x))), where the second limit is of the quotient form 0/0. If that
doesn't work, try lim_x®a ( f(x)g(x)) = lim_x®a (g(x)/(1/f(x))), where the second limit is of the form ¥/¥.

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4. The Indeterminate Exponential Forms

Here we're dealing with the limit of the exponential ( f(x))^g(x) as x ® a. Thus, suppose f(x) > 0 for all x
near a except possibly at a itself.

If lim_x®a ( f(x))^g(x) is of the form 0⁰, then it's of an indeterminate form, because ln lim_x®a ( f(x))^g(x) =
lim_x®a ln ( f(x))^g(x) = lim_x®a (g(x) ln f(x)), which is of the indeterminate product form 0 . (– ¥).

Similarly, 1^¥, 1^{– ¥}, and ¥⁰ are indeterminate exponential forms (corresponding to indeterminate product
forms ¥ . 0, (– ¥) . 0, and 0 . ¥ of their natural logarithms, respectively). We group the indeterminate
exponential forms together here:

0⁰, 1^¥, 1^{– ¥}, ¥⁰.

Evaluation

For example:







To evaluate a limit of any one of such forms, we take its logarithm, we get an indeterminate product
form, which we transform into an indeterminate quotient form, for which we can use L'Hôpital's Rule.
Then we take the exponential of the limit to obtain the original limit.

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Problems & Solutions


1. Evaluate the following limits, employing L'Hôpital's Rule if appropriate.



Solution













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2. Prove this limit:



Solution



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3. Prove that the exponential function e^x grows more rapidly for large positive x than any monomial
function xⁿ does.

Solution

Let n be any positive integer. We have:



where we use L'Hôpital's Rule n times. Hence, the exponential function e^x grows more rapidly for large
positive x than any monomial function xⁿ does.

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