So we infer that.
For the infinitelimit, , the inferenceof the limit is not as obvious.The functionappearsto
approachthe value but doesso very slowly, as the followingtablesuggests.
This unpredictablesituationwill applyto manyotherfunctionsof the form.Hencewe needanothermethod
that will providea differenttool for analyzingfunctionsof the form.
L’Hospital’s Rule: Let functions and be differentiableat everynumberotherthan in someinterval,
with if If , or if and
then:
- as long as this latterlimit existsor is infinite.
- If and are differentiableat everynumber greaterthan somenumber , with then
as long as this latterlimit existsor is infinite.
Let’s look at applyingthe rule to someexamples.
Example2:
We will start by reconsideringthe previousexample, and verifythe followinglimitsusing
L’Hospital’s Rule:
Solution:
Since , L’Hospital’s Rule appliesand we have