009A Sample Final 2, Problem 1

Compute

(a)  $\lim _{x\rightarrow 4}{\frac {{\sqrt {x+5}}-3}{x-4}}$

(b)  $\lim _{x\rightarrow 0}{\frac {\sin ^{2}x}{3x}}$

(c)  $\lim _{x\rightarrow -\infty }{\frac {\sqrt {x^{2}+2}}{2x-1}}$

Foundations:
L'Hôpital's Rule
Suppose that  $\lim _{x\rightarrow \infty }f(x)$  and  $\lim _{x\rightarrow \infty }g(x)$  are both zero or both  $\pm \infty .$
If  $\lim _{x\rightarrow \infty }{\frac {f'(x)}{g'(x)}}$  is finite or  $\pm \infty ,$
then  $\lim _{x\rightarrow \infty }{\frac {f(x)}{g(x)}}\,=\,\lim _{x\rightarrow \infty }{\frac {f'(x)}{g'(x)}}.$

Solution:

(a)

Step 1:
We begin by noticing that we plug in  $x=4$  into
${\frac {{\sqrt {x+5}}-3}{x-4}},$
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 4} \frac{\sqrt{x+5}-3}{x-4}} & = & \displaystyle{\lim_{x\rightarrow 4} \frac{\sqrt{x+5}-3}{x-4}\frac{(\sqrt{x+5}+3)}{(\sqrt{x+5}+3)}}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow 4} \frac{(x+5)-9}{(x-4)(\sqrt{x+5}+3)}}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow 4} \frac{x-4}{(x-4)(\sqrt{x+5}+3)}}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow 4} \frac{1}{\sqrt{x+5}+3}}\\ &&\\ & = & \displaystyle{ \frac{1}{\sqrt{9}+3}}\\ &&\\ & = & \displaystyle{\frac{1}{6}.} \end{array}}

(b)

Step 1:
We proceed using L'Hôpital's Rule. So, 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^2 (x)}{3x}} & \overset{L'H}{=} & \displaystyle{\lim_{x\rightarrow 0}\frac{2\sin(x)\cos(x)}{3}.} \end{array}}

Step 2:
Now, 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=0}   to 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 \begin{array}{rcl} \displaystyle{\lim_{x\rightarrow 0} \frac{\sin^2 (x)}{3x}} & = & \displaystyle{\frac{2\sin(0)\cos(0)}{3}}\\ &&\\ & = & \displaystyle{\frac{2(0)(1)}{3}}\\ &&\\ & = & \displaystyle{0.} \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}{6}}
(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 0}
(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 \frac{-1}{2}}

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