009A Sample Midterm 2, Problem 1

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:
1. lim sinx/x
2. Left and right hand limit

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:

Step 2:

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)

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