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| − | <span class="exam">Find the following limits: | + | <span class="exam"> Find the following limits: |
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| − | ::<span class="exam">a) If <math>\lim _{x\rightarrow 3} \bigg(\frac{f(x)}{2x}+1\bigg)=2,</math> find <math>\lim _{x\rightarrow 3} f(x).</math>
| + | <span class="exam">(a) If <math style="vertical-align: -16px">\lim _{x\rightarrow 3} \bigg(\frac{f(x)}{2x}+1\bigg)=2,</math> find <math style="vertical-align: -13px">\lim _{x\rightarrow 3} f(x).</math> |
| − | ::<span class="exam">b) Find <math>\lim _{x\rightarrow 0} \frac{\tan(4x)}{\sin(6x)}. </math>
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| − | ::<span class="exam">c) Evaluate <math>\lim _{x\rightarrow \infty} \frac{-2x^3-2x+3}{3x^3+3x^2-3}. </math>
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| | + | <span class="exam">(b) Find <math style="vertical-align: -19px">\lim _{x\rightarrow 0} \frac{\tan(4x)}{\sin(6x)}. </math> |
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| + | <span class="exam">(c) Evaluate <math style="vertical-align: -16px">\lim _{x\rightarrow \infty} \frac{-2x^3-2x+3}{3x^3+3x^2-3}. </math> |
| − | !Foundations:
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| − | |'''1.''' Linearity rules of limits
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| − | |'''2.''' lim sin(x)/x
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| − | '''Solution:''' | + | [[009A Sample Midterm 3, Problem 1 Detailed Solution|'''<u>Detailed Solution with Background Information</u>''']] |
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| − | '''(a)'''
| + | [[File:9ASM3P1.jpg|600px|thumb|center]] |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 1:
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| − | |First, we have
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| − | | <math>\begin{array}{rcl}
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| − | \displaystyle{2} & = & \displaystyle{\lim_{x\rightarrow 3} \bigg(\frac{f(x)}{2x}+1\bigg)}\\
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| − | &&\\
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| − | & = & \displaystyle{\lim_{x\rightarrow 3} \frac{f(x)}{2x}+\lim_{x\rightarrow 3} 1}\\
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| − | &&\\
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| − | & = & \displaystyle{\lim_{x\rightarrow 3} \frac{f(x)}{2x}+1.}
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| − | \end{array}</math>
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| − | |Therefore,
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| − | | <math>\lim_{x\rightarrow 3} \frac{f(x)}{2x}=1.</math> | |
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 2:
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| − | '''(b)'''
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| − | !Step 1:
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 2:
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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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| − | !Step 2:
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Final Answer:
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| − | |'''(a)'''
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| − | |'''(b)'''
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| − | |'''(c)'''
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| | [[009A_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']] | | [[009A_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']] |
Find the following limits:
(a) If 
find 
(b) Find 
(c) Evaluate 
Detailed Solution with Background Information
Return to Sample Exam
