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

Revision as of 09:12, 13 March 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) }

Foundations:
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 x}{x}=1}

Solution:

(a)

Step 1:
We begin by noticing that we plug in 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 x=2} into
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 \frac{\sqrt{x^2+12}-4}{x-2},}
we get 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 \frac{0}{0}.}

Step 2:
Now, we multiply the numerator and denominator by the conjugate of the numerator.
Hence, we have
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 \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}}

(b)

Step 1:

First, we write

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 \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}}

Step 2:

Now, we have

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 \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}}

(c)

Step 1:
We begin by looking at the graph of 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 y=\tan(x),}
which is displayed below.
(Insert graph)

Step 2:
We are taking a left hand limit. So, we approach 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 x=\frac{\pi}{2}} from the left.
If we look at the graph from the left of $x={\frac {\pi }{2}}$ and go towards 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 \frac{\pi}{2},}
we see 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 \tan(x)} goes to 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 \infty.}
Therefore,
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)=\infty.}

Final Answer:
(a) 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 \frac{1}{2}}
(b) 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 \frac{3}{7}}
(c) 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 \infty}

Return to Sample Exam

@@ Line 102: / Line 102: @@
 |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>
+|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>
+|&nbsp; &nbsp; &nbsp; &nbsp; <math>\lim _{x\rightarrow (\frac{\pi}{2})^-} \tan(x)=\infty.</math>
 |}
@@ Line 117: / Line 117: @@
 |&nbsp; &nbsp; '''(b)''' &nbsp; &nbsp; <math>\frac{3}{7}</math>
 |-
-|&nbsp; &nbsp; '''(c)''' &nbsp; &nbsp; <math>+\infty</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 09:12, 13 March 2017

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