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| | <span class="exam">Evaluate the following limits. | | <span class="exam">Evaluate the following limits. |
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| − | <span class="exam">(a) Find <math style="vertical-align: -14px">\lim _{x\rightarrow 2} \frac{\sqrt{x^2+12}-4}{x-2}</math> | + | <span class="exam">(a) Find <math style="vertical-align: -14px">\lim _{x\rightarrow 2} \frac{\sqrt{x^2+12}-4}{x-2}</math> |
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| − | <span class="exam">(b) Find <math style="vertical-align: -19px">\lim _{x\rightarrow 0} \frac{\sin(3x)}{\sin(7x)} </math> | + | <span class="exam">(b) Find <math style="vertical-align: -19px">\lim _{x\rightarrow 0} \frac{\sin(3x)}{\sin(7x)} </math> |
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| − | <span class="exam">(c) Evaluate <math style="vertical-align: -20px">\lim _{x\rightarrow (\frac{\pi}{2})^-} \tan(x) </math> | + | <span class="exam">(c) Evaluate <math style="vertical-align: -20px">\lim _{x\rightarrow (\frac{\pi}{2})^-} \tan(x) </math> |
| | + | <hr> |
| | + | [[009A Sample Midterm 2, Problem 1 Solution|'''<u>Solution</u>''']] |
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| + | [[009A Sample Midterm 2, Problem 1 Detailed Solution|'''<u>Detailed Solution</u>''']] |
| − | !Foundations:
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| − | |-
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| − | |<math>\lim_{x\rightarrow 0} \frac{\sin x}{x}=1</math> | |
| − | |}
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| − | '''Solution:'''
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| − |
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| − | '''(a)'''
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 1:
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| − | |-
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| − | ||We begin by noticing that we plug in <math>x=2</math> into
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| − | |-
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| − | | <math>\frac{\sqrt{x^2+12}-4}{x-2},</math>
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| − | |-
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| − | |we get <math>\frac{0}{0}.</math>
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| − | |}
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| − |
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 2:
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| − | |-
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| − | |Now, we multiply the numerator and denominator by the conjugate of the numerator.
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| − | |-
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| − | |Hence, we have
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| − | |-
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| − | | <math>\begin{array}{rcl}
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| − | \displaystyle{\lim _{x\rightarrow 2} \frac{\sqrt{x^2+12}-4}{x-2}} & = & \displaystyle{\lim_{x\rightarrow 2} \frac{(\sqrt{x^2+12}-4)}{(x-2)}\frac{(\sqrt{x^2+12}+4)}{(\sqrt{x^2+12}+4)}}\\
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| − | &&\\
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| − | & = & \displaystyle{\lim_{x\rightarrow 2} \frac{(x^2+12)-16}{(x-2)(\sqrt{x^2+12}+4)}}\\
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| − | &&\\
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| − | & = & \displaystyle{\lim_{x\rightarrow 2} \frac{x^2-4}{(x-2)(\sqrt{x^2+12}+4)}}\\
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| − | &&\\
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| − | & = & \displaystyle{\lim_{x\rightarrow 2} \frac{(x-2)(x+2)}{(x-2)(\sqrt{x^2+12}+4)}}\\
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| − | &&\\
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| − | & = & \displaystyle{\lim_{x\rightarrow 2} \frac{x+2}{\sqrt{x^2+12}+4}}\\
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| − | &&\\
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| − | & = & \displaystyle{\frac{4}{8}}\\
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| − | &&\\
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| − | & = & \displaystyle{\frac{1}{2}.}
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| − | \end{array}</math>
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| − | |}
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| − |
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| − | '''(b)'''
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 1:
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| − | |-
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| − | |First, we write
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| − | |-
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| − | | <math>\begin{array}{rcl}
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| − | \displaystyle{\lim_{x\rightarrow 0} \frac{\sin(3x)}{\sin(7x)}} & = & \displaystyle{\lim_{x\rightarrow 0} \frac{\sin(3x)}{x} \frac{x}{\sin(7x)}}\\
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| − | &&\\
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| − | & = & \displaystyle{\lim_{x\rightarrow 0} \frac{3}{7} \frac{\sin(3x)}{3x}\frac{7x}{\sin(7x)}}\\
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| − | &&\\
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| − | & = & \displaystyle{\frac{3}{7}\lim_{x\rightarrow 0} \frac{\sin(3x)}{3x}\frac{7x}{\sin(7x)}.}
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| − | \end{array}</math>
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| − | |}
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| − |
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 2:
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| − | |-
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| − | |Now, we have
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| − | |-
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| − | |
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| − | <math>\begin{array}{rcl}
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| − | \displaystyle{\lim_{x\rightarrow 0} \frac{\sin(3x)}{\sin(7x)}} & = & \displaystyle{\frac{3}{7}\lim_{x\rightarrow 0} \frac{\sin(3x)}{3x}\frac{7x}{\sin(7x)}}\\
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| − | &&\\
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| − | & = & \displaystyle{\frac{3}{7}\bigg(\lim_{x\rightarrow 0} \frac{\sin(3x)}{3x}\bigg)\bigg(\lim_{x\rightarrow 0} \frac{7x}{\sin(7x)}\bigg)}\\
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| − | &&\\
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| − | & = & \displaystyle{\frac{3}{7} (1)(1)}\\
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| − | &&\\
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| − | & = & \displaystyle{\frac{3}{7}.}
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| − | \end{array}</math>
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| − | |}
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| − |
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| − | '''(c)'''
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 1:
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| − | |-
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| − | |We begin by looking at the graph of <math>y=\tan(x),</math>
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| − | |-
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| − | |which is displayed below.
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| − | |-
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| − | |(Insert graph)
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| − | |}
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| − |
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 2:
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| − | |-
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| − | |We are taking a left hand limit. So, we approach <math>x=\frac{\pi}{2}</math> from the left.
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| − | |-
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| − | |If we look at the graph from the left of <math>x=\frac{\pi}{2}</math> and go towards <math>\frac{\pi}{2},</math>
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| − | |-
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| − | |we see that <math>\tan(x)</math> goes to <math>+\infty.</math>
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| − | |-
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| − | |Therefore,
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| − | |-
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| − | | <math>\lim _{x\rightarrow (\frac{\pi}{2})^-} \tan(x)=+\infty.</math>
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| − | |}
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| − |
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Final Answer:
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| − | |-
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| − | | '''(a)''' <math>\frac{1}{2}</math>
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| − | |-
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| − | | '''(b)''' <math>\frac{3}{7}</math>
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| − | |-
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| − | | '''(c)''' <math>+\infty</math>
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| − | |}
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| | [[009A_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']] | | [[009A_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']] |
