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

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(Created page with "<span class="exam">In each part, compute the limit. If the limit is infinite, be sure to specify positive or negative infinity. <span class="exam">a) <math style="vertical-al...")
 
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<span class="exam">In each part, compute the limit. If the limit is infinite, be sure to specify positive or negative infinity.
 
<span class="exam">In each part, compute the limit. If the limit is infinite, be sure to specify positive or negative infinity.
  
<span class="exam">a) <math style="vertical-align: -14px">\lim_{x\rightarrow -3} \frac{x^3-9x}{6+2x}</math>
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::<span class="exam">a) <math style="vertical-align: -14px">\lim_{x\rightarrow -3} \frac{x^3-9x}{6+2x}</math>
  
<span class="exam">b) <math style="vertical-align: -14px">\lim_{x\rightarrow 0^+} \frac{\sin (2x)}{x^2}</math>
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::<span class="exam">b) <math style="vertical-align: -14px">\lim_{x\rightarrow 0^+} \frac{\sin (2x)}{x^2}</math>
  
<span class="exam">c) <math style="vertical-align: -14px">\lim_{x\rightarrow -\infty} \frac{3x}{\sqrt{4x^2+x+5}}</math>
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::<span class="exam">c) <math style="vertical-align: -14px">\lim_{x\rightarrow -\infty} \frac{3x}{\sqrt{4x^2+x+5}}</math>
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"

Revision as of 11:57, 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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