Difference between revisions of "009A Sample Midterm 2, Problem 1"

Revision as of 09:01, 3 November 2017

Evaluate 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} \frac{\sqrt{x^2+12}-4}{x-2}}

(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(3x)}{\sin(7x)} }

(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 (\frac{\pi}{2})^-} \tan(x) }

(insert picture of handwritten solution)

Detailed Solution for this Problem

Return to Sample Exam

@@ Line 6: / Line 6: @@
 <span class="exam">(c) Evaluate &nbsp;<math style="vertical-align: -20px">\lim _{x\rightarrow (\frac{\pi}{2})^-} \tan(x) </math>
+<hr>
+(insert picture of handwritten solution)
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+[[009A Sample Midterm 2, Problem 1 Detailed Solution|'''<u>Detailed Solution for this Problem</u>''']]
-!Foundations: &nbsp;
-|-
-|Recall
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp;<math>\lim_{x\rightarrow 0} \frac{\sin x}{x}=1</math>
-|}
-'''Solution:'''
-'''(a)'''
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 1: &nbsp;
-|-
-|We begin by noticing that we plug in &nbsp;<math style="vertical-align: 0px">x=2</math>&nbsp; into
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp; <math>\frac{\sqrt{x^2+12}-4}{x-2},</math>
-|-
-|we get &nbsp; <math style="vertical-align: -12px">\frac{0}{0}.</math>
-|}
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 2: &nbsp;
-|-
-|Now, we multiply the numerator and denominator by the conjugate of the numerator.
-|-
-|Hence, we have
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
-\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)}}\\
-&&\\
-& = & \displaystyle{\lim_{x\rightarrow 2} \frac{(x^2+12)-16}{(x-2)(\sqrt{x^2+12}+4)}}\\
-&&\\
-& = & \displaystyle{\lim_{x\rightarrow 2} \frac{x^2-4}{(x-2)(\sqrt{x^2+12}+4)}}\\
-&&\\
-& = & \displaystyle{\lim_{x\rightarrow 2} \frac{(x-2)(x+2)}{(x-2)(\sqrt{x^2+12}+4)}}\\
-&&\\
-& = & \displaystyle{\lim_{x\rightarrow 2} \frac{x+2}{\sqrt{x^2+12}+4}}\\
-&&\\
-& = & \displaystyle{\frac{4}{8}}\\
-&&\\
-& = & \displaystyle{\frac{1}{2}.}
-\end{array}</math>
-|}
-'''(b)'''
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 1: &nbsp;
-|-
-|First, we write
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
-\displaystyle{\lim_{x\rightarrow 0} \frac{\sin(3x)}{\sin(7x)}} & = & \displaystyle{\lim_{x\rightarrow 0} \frac{\sin(3x)}{x} \frac{x}{\sin(7x)}}\\
-&&\\
-& = & \displaystyle{\lim_{x\rightarrow 0} \frac{3}{7} \frac{\sin(3x)}{3x}\frac{7x}{\sin(7x)}}\\
-&&\\
-& = & \displaystyle{\frac{3}{7}\lim_{x\rightarrow 0} \frac{\sin(3x)}{3x}\frac{7x}{\sin(7x)}.}
-\end{array}</math>
-|}
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 2: &nbsp;
-|-
-|Now, we have
-|-
-|
-&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
-\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)}}\\
-&&\\
-& = & \displaystyle{\frac{3}{7}\bigg(\lim_{x\rightarrow 0} \frac{\sin(3x)}{3x}\bigg)\bigg(\lim_{x\rightarrow 0} \frac{7x}{\sin(7x)}\bigg)}\\
-&&\\
-& = & \displaystyle{\frac{3}{7} (1)(1)}\\
-&&\\
-& = & \displaystyle{\frac{3}{7}.}
-\end{array}</math>
-|}
-'''(c)'''
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 1: &nbsp;
-|-
-|We begin by looking at the graph of &nbsp;<math style="vertical-align: -5px">y=\tan(x),</math>
-|-
-|which is displayed below.
-|-
-|(Insert graph)
-|}
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 2: &nbsp;
-|-
-|We are taking a left hand limit. So, we approach &nbsp;<math style="vertical-align: -13px">x=\frac{\pi}{2}</math> &nbsp; from the left.
-|-
-|If we look at the graph from the left of &nbsp;<math style="vertical-align: -13px">x=\frac{\pi}{2}</math> &nbsp; and go towards &nbsp; <math style="vertical-align: -13px">\frac{\pi}{2},</math>
-|-
-|we see that &nbsp;<math style="vertical-align: -5px">\tan(x)</math> &nbsp; goes to &nbsp;<math style="vertical-align: -2px">\infty.</math>
-|-
-|Therefore,
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp; <math>\lim _{x\rightarrow (\frac{\pi}{2})^-} \tan(x)=\infty.</math>
-|}
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Final Answer: &nbsp;
-|-
-|&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp; <math>\frac{1}{2}</math>
-|-
-|&nbsp; &nbsp; '''(b)''' &nbsp; &nbsp; <math>\frac{3}{7}</math>
-|-
-|&nbsp; &nbsp; '''(c)''' &nbsp; &nbsp; <math>\infty</math>
-|}
 [[009A_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]

Difference between revisions of "009A Sample Midterm 2, Problem 1"

Revision as of 09:01, 3 November 2017

Navigation menu

Search