|
|
| Line 7: |
Line 7: |
| | <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> | | <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> |
| | | | |
| | + | <hr> |
| | | | |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| + | [[009A Sample Midterm 3, Problem 1 Detailed Solution|'''<u>Detailed Solution with Background Information</u>''']] |
| − | !Foundations:
| |
| − | |-
| |
| − | |'''1.''' If <math style="vertical-align: -13px">\lim_{x\rightarrow a} g(x)\neq 0,</math> we have | |
| − | |-
| |
| − | | <math>\lim_{x\rightarrow a} \frac{f(x)}{g(x)}=\frac{\displaystyle{\lim_{x\rightarrow a} f(x)}}{\displaystyle{\lim_{x\rightarrow a} g(x)}}.</math>
| |
| − | |-
| |
| − | |'''2.''' Recall
| |
| − | |-
| |
| − | | <math style="vertical-align: -15px">\lim_{x\rightarrow 0} \frac{\sin x}{x}=1</math>
| |
| − | |}
| |
| | | | |
| | + | [[File:9ASM3P1.jpg|600px|thumb|center]] |
| | | | |
| − | '''Solution:'''
| |
| − |
| |
| − | '''(a)'''
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 1:
| |
| − | |-
| |
| − | |First, we have
| |
| − | |-
| |
| − | | <math>\begin{array}{rcl}
| |
| − | \displaystyle{2} & = & \displaystyle{\lim_{x\rightarrow 3} \bigg(\frac{f(x)}{2x}+1\bigg)}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{\lim_{x\rightarrow 3} \frac{f(x)}{2x}+\lim_{x\rightarrow 3} 1}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{\lim_{x\rightarrow 3} \frac{f(x)}{2x}+1.}
| |
| − | \end{array}</math>
| |
| − | |-
| |
| − | |Therefore,
| |
| − | |-
| |
| − | | <math>\lim_{x\rightarrow 3} \frac{f(x)}{2x}=1.</math>
| |
| − | |}
| |
| − |
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 2:
| |
| − | |-
| |
| − | |Since <math style="vertical-align: -13px">\lim_{x\rightarrow 3} 2x=6\ne 0,</math> we have
| |
| − | |-
| |
| − | |
| |
| − | <math>\begin{array}{rcl}
| |
| − | \displaystyle{1} & = & \displaystyle{\lim_{x\rightarrow 3} \frac{f(x)}{2x}}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{\frac{\displaystyle{\lim_{x\rightarrow 3} f(x)}}{\displaystyle{\lim_{x\rightarrow 3} 2x}}}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{\frac{\displaystyle{\lim_{x\rightarrow 3} f(x)}}{6}.}
| |
| − | \end{array}</math>
| |
| − | |-
| |
| − | |Multiplying both sides by <math style="vertical-align: -5px">6,</math> we get
| |
| − | |-
| |
| − | | <math>\lim_{x\rightarrow 3} f(x)=6.</math>
| |
| − | |}
| |
| − |
| |
| − | '''(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{\tan(4x)}{\sin(6x)}} & = & \displaystyle{\lim_{x\rightarrow 0} \frac{\sin(4x)}{\cos(4x)} \frac{1}{\sin(6x)}}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{\lim_{x\rightarrow 0} \frac{4}{6} \frac{\sin(4x)}{4x}\frac{6x}{\sin(6x)}\frac{1}{\cos(4x)}}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{\frac{4}{6}\lim_{x\rightarrow 0} \frac{\sin(4x)}{4x}\frac{6x}{\sin(6x)}\frac{1}{\cos(4x)}.}
| |
| − | \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{\tan(4x)}{\sin(6x)}} & = & \displaystyle{\frac{4}{6}\lim_{x\rightarrow 0} \frac{\sin(4x)}{4x}\frac{6x}{\sin(6x)}\frac{1}{\cos(4x)}}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{\frac{4}{6}\bigg(\lim_{x\rightarrow 0} \frac{\sin(4x)}{4x}\bigg)\bigg(\lim_{x\rightarrow 0} \frac{6x}{\sin(6x)}\bigg)\bigg(\lim_{x\rightarrow 0} \frac{1}{\cos(4x)}\bigg)}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{\frac{4}{6} (1)(1)(1)}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{\frac{2}{3}.}
| |
| − | \end{array}</math>
| |
| − | |}
| |
| − |
| |
| − | '''(c)'''
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 1:
| |
| − | |-
| |
| − | |First, we have
| |
| − | |-
| |
| − | | <math>\begin{array}{rcl}
| |
| − | \displaystyle{\lim _{x\rightarrow \infty} \frac{-2x^3-2x+3}{3x^3+3x^2-3}} & = & \displaystyle{\lim _{x\rightarrow \infty} \frac{(-2x^3-2x+3)}{(3x^3+3x^2-3)} \frac{(\frac{1}{x^3})}{(\frac{1}{x^3})}}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{\lim_{x\rightarrow \infty} \frac{-2-\frac{2}{x^2}+\frac{3}{x^3}}{3+\frac{3}{x}-\frac{3}{x^3}}}.
| |
| − | \end{array}</math>
| |
| − | |}
| |
| − |
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 2:
| |
| − | |-
| |
| − | |Now, we use the properties of limits to get
| |
| − | |-
| |
| − | |
| |
| − | <math>\begin{array}{rcl}
| |
| − | \displaystyle{\lim _{x\rightarrow \infty} \frac{-2x^3-2x+3}{3x^3+3x^2-3}} & = & \displaystyle{\lim_{x\rightarrow \infty} \frac{-2-\frac{2}{x^2}+\frac{3}{x^3}}{3+\frac{3}{x}-\frac{3}{x^3}}}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{\frac{\displaystyle{\lim_{x\rightarrow \infty} \bigg(-2-\frac{2}{x^2}+\frac{3}{x^3}\bigg)}}{\displaystyle{\lim_{x\rightarrow \infty} \bigg(3+\frac{3}{x}-\frac{3}{x^3}\bigg)}}}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{\frac{\displaystyle{\lim_{x\rightarrow \infty} -2 +\lim_{x\rightarrow \infty} \frac{2}{x^2} +\lim_{x\rightarrow \infty} \frac{3}{x^3}}}{\displaystyle{\lim_{x\rightarrow \infty} 3+\lim_{x\rightarrow \infty} \frac{3}{x}-\lim_{x\rightarrow \infty}\frac{3}{x^3}}}} \\
| |
| − | &&\\
| |
| − | & = & \displaystyle{\frac{-2+0+0}{3+0+0}}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{-\frac{2}{3}.}
| |
| − | \end{array}</math>
| |
| − | |}
| |
| − |
| |
| − |
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Final Answer:
| |
| − | |-
| |
| − | | '''(a)''' <math>6</math>
| |
| − | |-
| |
| − | | '''(b)''' <math>\frac{2}{3}</math>
| |
| − | |-
| |
| − | | '''(c)''' <math>-\frac{2}{3}</math>
| |
| − | |}
| |
| | [[009A_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']] | | [[009A_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']] |
