007A Sample Final 2, Problem 8 Detailed Solution

Compute

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

(b) $\lim _{x\rightarrow 0}{\frac {\sin x}{\cos x-1}}$

(c) $\lim _{x\rightarrow 1}{\frac {x^{3}-1}{x^{10}-1}}$

Background Information:
L'Hôpital's Rule, Part 1
Let $\lim _{x\rightarrow c}f(x)=0$ and $\lim _{x\rightarrow c}g(x)=0,$ where $f$ and $g$ are differentiable functions
on an open interval $I$ containing $c,$ and $g'(x)\neq 0$ on $I$ except possibly at $c.$
Then, $\lim _{x\rightarrow c}{\frac {f(x)}{g(x)}}=\lim _{x\rightarrow c}{\frac {f'(x)}{g'(x)}}.$

Solution:

(a)

Step 1:

First, we have

{\begin{array}{rcl}\displaystyle {\lim _{x\rightarrow \infty }{\frac {x^{-1}+x}{1+{\sqrt {1+x}}}}}&=&\displaystyle {\lim _{x\rightarrow \infty }{\frac {{\frac {1}{x}}+x}{1+{\sqrt {1+x}}}}}\\&&\\&=&\displaystyle {\lim _{x\rightarrow \infty }{\frac {{\frac {1}{x}}+x}{1+{\sqrt {1+x}}}}\cdot {\frac {{\big (}{\frac {1}{\sqrt {x}}}{\big )}}{{\big (}{\frac {1}{\sqrt {x}}}{\big )}}}}\\&&\\&=&\displaystyle {\lim _{x\rightarrow \infty }{\frac {{\frac {1}{x^{3/2}}}+{\sqrt {x}}}{{\frac {1}{\sqrt {x}}}+{\sqrt {{\frac {1}{x}}+1}}}}.}\end{array}}

Step 2:

Now, we have

{\begin{array}{rcl}\displaystyle {\lim _{x\rightarrow \infty }{\frac {x^{-1}+x}{1+{\sqrt {1+x}}}}}&=&\displaystyle {\lim _{x\rightarrow \infty }{\frac {{\frac {1}{x^{3/2}}}+{\sqrt {x}}}{{\frac {1}{\sqrt {x}}}+{\sqrt {{\frac {1}{x}}+1}}}}}\\&&\\&=&\displaystyle {\frac {\displaystyle {\lim _{x\rightarrow \infty }}{\bigg (}{\frac {1}{x^{3/2}}}+{\sqrt {x}}{\bigg )}}{\displaystyle {\lim _{x\rightarrow \infty }}{\bigg (}{\frac {1}{\sqrt {x}}}+{\sqrt {{\frac {1}{x}}+1}}{\bigg )}}}\\&&\\&=&\displaystyle {\frac {\displaystyle {\lim _{x\rightarrow \infty }}{\frac {1}{x^{3/2}}}+\displaystyle {\lim _{x\rightarrow \infty }}{\sqrt {x}}}{\displaystyle {\lim _{x\rightarrow \infty }}{\frac {1}{\sqrt {x}}}+\displaystyle {\lim _{x\rightarrow \infty }}{\sqrt {{\frac {1}{x}}+1}}}}\\&&\\&=&\displaystyle {\frac {0+\displaystyle {\lim _{x\rightarrow \infty }}{\sqrt {x}}}{0+1}}\\&&\\&=&\displaystyle {\infty .}\end{array}}

(b)

Step 1:

First, we write

{\begin{array}{rcl}\displaystyle {\lim _{x\rightarrow 0}{\frac {\sin x}{\cos x-1}}}&=&\displaystyle {\lim _{x\rightarrow 0}{\frac {\sin x}{\cos x-1}}{\frac {(\cos x+1)}{(\cos x+1)}}}\\&&\\&=&\displaystyle {\lim _{x\rightarrow 0}{\frac {\sin x(\cos x+1)}{\cos ^{2}x-1}}}\\&&\\&=&\displaystyle {\lim _{x\rightarrow 0}{\frac {\sin x(\cos x+1)}{-\sin ^{2}x}}}\\&&\\&=&\displaystyle {\lim _{x\rightarrow 0}{\frac {\cos x+1}{-\sin x}}.}\end{array}}

Step 2:
Now, we have
${\begin{array}{rcl}\displaystyle {\lim _{x\rightarrow 0^{+}}{\frac {\sin x}{\cos x-1}}}&=&\displaystyle {\lim _{x\rightarrow 0^{+}}{\frac {\cos x+1}{-\sin x}}}\\&&\\&=&\displaystyle {-\infty }\end{array}}$
and
${\begin{array}{rcl}\displaystyle {\lim _{x\rightarrow 0^{-}}{\frac {\sin x}{\cos x-1}}}&=&\displaystyle {\lim _{x\rightarrow 0^{-}}{\frac {\cos x+1}{-\sin x}}}\\&&\\&=&\displaystyle {\infty .}\end{array}}$
Therefore,
$\lim _{x\rightarrow 0}{\frac {\sin x}{\cos x-1}}={\text{DNE}}.$

(c)

Step 1:
We proceed using L'Hôpital's Rule. So, we have
${\begin{array}{rcl}\displaystyle {\lim _{x\rightarrow 1}{\frac {x^{3}-1}{x^{10}-1}}}&{\overset {L'H}{=}}&\displaystyle {\lim _{x\rightarrow 1}{\frac {3x^{2}}{10x^{9}}}.}\end{array}}$

Step 2:
Now, we have
${\begin{array}{rcl}\displaystyle {\lim _{x\rightarrow 1}{\frac {x^{3}-1}{x^{10}-1}}}&{\overset {L'H}{=}}&\displaystyle {\lim _{x\rightarrow 1}{\frac {3x^{2}}{10x^{9}}}}\\&&\\&=&\displaystyle {\frac {3(1)^{2}}{10(1)^{9}}}\\&&\\&=&\displaystyle {{\frac {3}{10}}.}\end{array}}$

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