009A Sample Final 2, Problem 1

Compute

(a)  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lim _{x\rightarrow 4}{\frac {{\sqrt {x+5}}-3}{x-4}}}

(b)  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lim _{x\rightarrow 0}{\frac {\sin ^{2}x}{3x}}}

(c)  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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   ${\frac {0}{0}}.$

Step 2:
Now, we multiply the numerator and denominator by the conjugate of the numerator.
Hence, we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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  $x=0$  to get
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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:
First, we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {\lim _{x\rightarrow -\infty }{\frac {\sqrt {x^{2}+2}}{2x-1}}}&=&\displaystyle {\lim _{x\rightarrow -\infty }{\frac {\sqrt {x^{2}(1+{\frac {2}{x^{2}}})}}{2x-1}}}\\&&\\&=&\displaystyle {\lim _{x\rightarrow -\infty }{\frac {|x|{\sqrt {1+{\frac {2}{x^{2}}}}}}{2x-1}}}\\&&\\&=&\displaystyle {\lim _{x\rightarrow -\infty }{\frac {-x{\sqrt {1+{\frac {2}{x^{2}}}}}}{x(2-{\frac {1}{x}})}}}\\&&\\&=&\displaystyle {\lim _{x\rightarrow -\infty }{\frac {-1{\sqrt {1+{\frac {2}{x^{2}}}}}}{(2-{\frac {1}{x}})}}.}\end{array}}}

Step 2:
Now,
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {\lim _{x\rightarrow -\infty }{\frac {\sqrt {x^{2}+2}}{2x-1}}}&=&\displaystyle {\lim _{x\rightarrow -\infty }{\frac {-1{\sqrt {1+{\frac {2}{x^{2}}}}}}{(2-{\frac {1}{x}})}}}\\&&\\&=&\displaystyle {\frac {-{\sqrt {1+0}}}{(2-0)}}\\&&\\&=&\displaystyle {-{\frac {1}{2}}.}\end{array}}}

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}}

Return to Sample Exam