009A Sample Midterm 2, Problem 1

Evaluate the following limits.

(a) Find  $\lim _{x\rightarrow 2}{\frac {{\sqrt {x^{2}+12}}-4}{x-2}}$

(b) Find  $\lim _{x\rightarrow 0}{\frac {\sin(3x)}{\sin(7x)}}$

(c) Evaluate  $\lim _{x\rightarrow ({\frac {\pi }{2}})^{-}}\tan(x)$

Foundations:
Recall
$\lim _{x\rightarrow 0}{\frac {\sin x}{x}}=1$

Solution:

(a)

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

Step 2:
Now, we multiply the numerator and denominator by the conjugate of the numerator.
Hence, we have
${\begin{array}{rcl}\displaystyle {\lim _{x\rightarrow 2}{\frac {{\sqrt {x^{2}+12}}-4}{x-2}}}&=&\displaystyle {\lim _{x\rightarrow 2}{\frac {({\sqrt {x^{2}+12}}-4)}{(x-2)}}{\frac {({\sqrt {x^{2}+12}}+4)}{({\sqrt {x^{2}+12}}+4)}}}\\&&\\&=&\displaystyle {\lim _{x\rightarrow 2}{\frac {(x^{2}+12)-16}{(x-2)({\sqrt {x^{2}+12}}+4)}}}\\&&\\&=&\displaystyle {\lim _{x\rightarrow 2}{\frac {x^{2}-4}{(x-2)({\sqrt {x^{2}+12}}+4)}}}\\&&\\&=&\displaystyle {\lim _{x\rightarrow 2}{\frac {(x-2)(x+2)}{(x-2)({\sqrt {x^{2}+12}}+4)}}}\\&&\\&=&\displaystyle {\lim _{x\rightarrow 2}{\frac {x+2}{{\sqrt {x^{2}+12}}+4}}}\\&&\\&=&\displaystyle {\frac {4}{8}}\\&&\\&=&\displaystyle {{\frac {1}{2}}.}\end{array}}$

(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  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}}   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}

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