In this section we compute limits using L’Hopital’s Rule which requires ourknowledge of derivatives.
L’Hopital’s Rule
L’Hopital’s Rule uses the derivative to help us find limits involving indeterminateforms. The main indeterminate forms we will discuss are and . We begin with thefractional forms.
provided the latter limit exists. The statement is also true for one-sided limits and if‘’ is replaced by or .
The indeterminate form
Plugging in the terminal value, , yields the indeterminate form , so L’Hopital’s ruleapplies.
We have
example 2 Compute the limit:
Plugging in the terminal value, , yields the indeterminate form , so L’Hopital’s ruleapplies. We have
If you got , use L’Hopital’s Rule
Take the derivative of the numerator and denominator separately
Plugging in the terminal value, , gives the indeterminate form , so we can useL’Hopital’s Rule:
(problem 3c) Let and be non-zero constants. Compute
Check for fractional indeterminate form.
Sometimes we have to use L’Hopital’s Rule more than once.
Plugging in the terminal value, , yields the indeterminate form , so L’Hopital’s ruleapplies. We have Applying L’Hopital’s Rule again gives Hence
If you got , use L’Hopital’s Rule
Take the derivative of the numerator and denominator separately
If you got , use L’Hopital’s Rule
Take the derivative of the numerator and denominator separately
Compute the limit of the new fraction
example 5 Compute the limit:
Plugging in the terminal value, , yields the indeterminate form , so L’Hopital’s ruleapplies. We have Applying L’Hopital’s Rule again gives We need to applyL’Hopital’s Rule again, but first, the numerator is complicated and so we take asimplifying step before applying the rule.
We next consider problems of the form . These are handled the same way as the caseabove.
The case.
As approaches we get the indeterminate form so L’Hopital’s Rule applies. We have Applying L’Hopital again, we get Hence . This limit can be generalized as follows: for any exponent . This general result comes from using L’Hopital’s Rule times,yielding where . The interpretation of this limit is that the exponential function grows faster than any power of as .
If you got , use L’Hopital’s Rule
Take the derivative of the numerator and denominator separately
example 7 Compute the limit: . As we get , so L’Hopital’s Rule applies. We have: which simplifies to Hence, . The interpretation of this limit is that goes to fasterthan as .
The case
L’Hopital’s Rule requires a fractional indeterminate form such as or , but wecan use it to handle other indeterminate forms by rewriting expressions asfractions.
Examples of the case.
example 8 Compute the limit: .
As we get which is an indeterminate form, but L’Hopital’s Rule does not apply inthis situation. We must rewrite the problem as a fraction, in the following way: Notice that this is equivalent to the original problem since Also note that as . Now,we can use L’Hopital’s Rule because We get which simplifies to Hence,
Here is a detailed, lecture style video on L’Hopital’s Rule:
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