Difference between revisions of "009A Sample Final 1, Problem 1"

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|'''L'Hôpital's Rule'''  
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::'''L'Hôpital's Rule'''  
 
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|Suppose that <math style="vertical-align: -11px">\lim_{x\rightarrow \infty} f(x)</math>&thinsp; and <math style="vertical-align: -11px">\lim_{x\rightarrow \infty} g(x)</math>&thinsp; are both zero or both <math style="vertical-align: -1px">\pm \infty .</math>
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::Suppose that <math style="vertical-align: -11px">\lim_{x\rightarrow \infty} f(x)</math>&thinsp; and <math style="vertical-align: -11px">\lim_{x\rightarrow \infty} g(x)</math>&thinsp; are both zero or both <math style="vertical-align: -1px">\pm \infty .</math>
 
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Revision as of 12:04, 18 April 2016

In each part, compute the limit. If the limit is infinite, be sure to specify positive or negative infinity.

a)
b)
c)
Foundations:  
Recall:
L'Hôpital's Rule
Suppose that   and   are both zero or both
If   is finite or 
then

Solution:

(a)

Step 1:  
We begin by factoring the numerator. We have
So, we can cancel   in the numerator and denominator. Thus, we have
Step 2:  
Now, we can just plug in   to get

(b)

Step 1:  
We proceed using L'Hôpital's Rule. So, we have
Step 2:  
This limit is 

(c)

Step 1:  
We have
Since we are looking at the limit as goes to negative infinity, we have
So, we have
Step 2:  
We simplify to get
So, we have
Final Answer:  
(a)
(b)
(c)

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