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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) Find <math>\lim _{x\rightarrow 2} g(x),</math> provided that <math>\lim _{x\rightarrow 2} \bigg[\frac{4-g(x)}{x}\bigg]=5</math>
| + | <span class="exam">(a) Find <math style="vertical-align: -13px">\lim _{x\rightarrow 2} g(x),</math> provided that <math style="vertical-align: -15px">\lim _{x\rightarrow 2} \bigg[\frac{4-g(x)}{x}\bigg]=5.</math> |
| − | ::<span class="exam">b) Find <math>\lim _{x\rightarrow 0} \frac{\sin(4x)}{5x} </math>
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| − | ::<span class="exam">c) Evaluate <math>\lim _{x\rightarrow -3^+} \frac{x}{x^2-9} </math>
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| | + | <span class="exam">(b) Find <math style="vertical-align: -14px">\lim _{x\rightarrow 0} \frac{\sin(4x)}{5x} </math> |
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| + | <span class="exam">(c) Evaluate <math style="vertical-align: -14px">\lim _{x\rightarrow -3^+} \frac{x}{x^2-9} </math> |
| − | !Foundations:
| + | <hr> |
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| + | [[009A Sample Midterm 1, Problem 1 Detailed Solution|'''<u>Detailed Solution with Background Information</u>''']] |
| − | | '''1.''' Linearity rules of limits
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| − | | '''2.''' Limit sin(x)/x
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| − | |'''3.''' Left and right hand limits | |
| − | |}
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| − | '''Solution:'''
| + | [[File:9ASM1P1.jpg|600px|thumb|center]] |
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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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| − | |Since <math>\lim_{x\rightarrow 2} x =2\ne 0,</math>
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| − | |we have
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| − | | <math>\begin{array}{rcl}
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| − | \displaystyle{5} & = & \displaystyle{\lim _{x\rightarrow 2} \bigg[\frac{4-g(x)}{x}\bigg]}\\
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| − | &&\\
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| − | & = & \displaystyle{\frac{\lim_{x\rightarrow 2} (4-g(x))}{\lim_{x\rightarrow 2} x}}\\
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| − | &&\\
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| − | & = & \displaystyle{\frac{\lim_{x\rightarrow 2} (4-g(x))}{2}.}
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| − | \end{array}</math>
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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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| − | |If we multiply both sides of the last equation by <math>2,</math> we get
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| − | | <math>10=\lim_{x\rightarrow 2} (4-g(x)).</math>
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| − | |Now, using linearity properties of limits, we have
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| − | | <math>\begin{array}{rcl}
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| − | \displaystyle{10} & = & \displaystyle{\lim_{x\rightarrow 2} 4 -\lim_{x\rightarrow 2}g(x)}\\
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| − | &&\\
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| − | & = & \displaystyle{4-\lim_{x\rightarrow 2} g(x).}\\
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| − | \end{array}</math>
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| − | |}
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 3:
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| − | |Solving for <math>\lim_{x\rightarrow 2} g(x)</math> in the last equation,
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| − | |we get
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| − | <math> \lim_{x\rightarrow 2} g(x)=-6.</math>
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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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| − | |First, we write
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| − | | <math>\lim_{x\rightarrow 0} \frac{\sin(4x)}{5x}=\lim_{x\rightarrow 0} \frac{4}{5} \bigg(\frac{\sin(4x)}{4x}\bigg).</math>
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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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| − | |Now, we have
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| − | | <math>\begin{array}{rcl}
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| − | \displaystyle{\lim_{x\rightarrow 0} \frac{\sin(4x)}{5x}} & = & \displaystyle{\frac{4}{5}\lim_{x\rightarrow 0} \frac{\sin(4x)}{4x}}\\
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| − | &&\\
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| − | & = & \displaystyle{\frac{4}{5}(1)}\\
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| − | &&\\
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| − | & = & \displaystyle{\frac{4}{5}.}
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| − | \end{array}</math>
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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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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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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)''' <math> \lim_{x\rightarrow 2} g(x)=-6</math>
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| − | |'''(b)'''
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| − | |'''(c)'''
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| − | |}
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| | [[009A_Sample_Midterm_1|'''<u>Return to Sample Exam</u>''']] | | [[009A_Sample_Midterm_1|'''<u>Return to Sample Exam</u>''']] |
Find the following limits:
(a) Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim _{x\rightarrow 2} g(x),}
provided that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim _{x\rightarrow 2} \bigg[\frac{4-g(x)}{x}\bigg]=5.}
(b) Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim _{x\rightarrow 0} \frac{\sin(4x)}{5x} }
(c) Evaluate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim _{x\rightarrow -3^+} \frac{x}{x^2-9} }
Detailed Solution with Background Information
Return to Sample Exam
