|
|
| (3 intermediate revisions by the same user not shown) |
| Line 6: |
Line 6: |
| | | | |
| | <span class="exam">(c) <math style="vertical-align: -15px">\lim_{x\rightarrow 1} \frac{x^3-1}{x^{10}-1}</math> | | <span class="exam">(c) <math style="vertical-align: -15px">\lim_{x\rightarrow 1} \frac{x^3-1}{x^{10}-1}</math> |
| | + | <hr> |
| | + | [[009A Sample Final 2, Problem 8 Solution|'''<u>Solution</u>''']] |
| | | | |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Foundations:
| |
| − | |-
| |
| − | |<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> | + | [[009A Sample Final 2, Problem 8 Detailed Solution|'''<u>Detailed Solution</u>''']] |
| | | | |
| − | <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>
| |
| − | |}
| |
| − |
| |
| − |
| |
| − | '''Solution:'''
| |
| − |
| |
| − | '''(a)'''
| |
| − |
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 1:
| |
| − | |-
| |
| − | |
| |
| − | |-
| |
| − | |
| |
| − | |-
| |
| − | |
| |
| − | |-
| |
| − | |
| |
| − | |}
| |
| − |
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 2:
| |
| − | |-
| |
| − | |
| |
| − | |-
| |
| − | |
| |
| − | |}
| |
| − |
| |
| − | '''(b)'''
| |
| − |
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 1:
| |
| − | |-
| |
| − | |First, we write
| |
| − | |-
| |
| − | | <math>\begin{array}{rcl}
| |
| − | \displaystyle{\lim_{x\rightarrow 0} \frac{\sin x}{\cos x-1}} & = & \displaystyle{\lim_{x\rightarrow 0} \frac{\sin x}{\cos x-1}\frac{(\cos x+1)}{(\cos x+1)}}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{\lim_{x\rightarrow 0} \frac{\sin x (\cos x+1)}{\cos^2x-1}}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{\lim_{x\rightarrow 0} \frac{\sin x(\cos x+1)}{-\sin^2 x}}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{\lim_{x\rightarrow 0} \frac{\cos x+1}{-\sin x}.}
| |
| − | \end{array}</math>
| |
| − | |}
| |
| − |
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 2:
| |
| − | |-
| |
| − | |Now, we have
| |
| − | |-
| |
| − | | <math>\begin{array}{rcl}
| |
| − | \displaystyle{\lim_{x\rightarrow 0^+} \frac{\sin x}{\cos x-1}} & = & \displaystyle{\lim_{x\rightarrow 0^+} \frac{\cos x+1}{-\sin x}}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{-\infty}
| |
| − | \end{array}</math>
| |
| − | |-
| |
| − | |and
| |
| − | |-
| |
| − | | <math>\begin{array}{rcl}
| |
| − | \displaystyle{\lim_{x\rightarrow 0^-} \frac{\sin x}{\cos x-1}} & = & \displaystyle{\lim_{x\rightarrow 0^-} \frac{\cos x+1}{-\sin x}}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{\infty.}
| |
| − | \end{array}</math>
| |
| − | |-
| |
| − | |Therefore,
| |
| − | |-
| |
| − | | <math>\lim_{x\rightarrow 0} \frac{\sin x}{\cos x-1}=\text{DNE}.</math>
| |
| − | |}
| |
| − |
| |
| − | '''(c)'''
| |
| − |
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 1:
| |
| − | |-
| |
| − | |We proceed using L'Hôpital's Rule. So, we have
| |
| − | |-
| |
| − | |
| |
| − | <math>\begin{array}{rcl}
| |
| − | \displaystyle{\lim_{x\rightarrow 1} \frac{x^3-1}{x^{10}-1}} & \overset{L'H}{=} & \displaystyle{\lim_{x\rightarrow 1}\frac{3x^2}{10x^9}.}
| |
| − | \end{array}</math>
| |
| − | |}
| |
| − |
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 2:
| |
| − | |-
| |
| − | |Now, we have
| |
| − | |-
| |
| − | | <math>\begin{array}{rcl}
| |
| − | \displaystyle{\lim_{x\rightarrow 1} \frac{x^3-1}{x^{10}-1}} & \overset{L'H}{=} & \displaystyle{\lim_{x\rightarrow 1}\frac{3x^2}{10x^9}}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{\frac{3(1)^2}{10(1)^9}}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{\frac{3}{10}.}
| |
| − | \end{array}</math>
| |
| − | |}
| |
| − |
| |
| − |
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Final Answer:
| |
| − | |-
| |
| − | | '''(a)'''
| |
| − | |-
| |
| − | | '''(b)''' <math>\text{DNE}</math>
| |
| − | |-
| |
| − | | '''(c)''' <math>\frac{3}{10}</math>
| |
| − | |}
| |
| | [[009A_Sample_Final_2|'''<u>Return to Sample Exam</u>''']] | | [[009A_Sample_Final_2|'''<u>Return to Sample Exam</u>''']] |
