CPM Homework Help : APCALC Problem 10-111

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10-111.

Evaluate each of the following limits.

$\lim\limits _ { x \rightarrow \infty } \frac { x ^ { 3 } } { e ^ { x } }$
Hint (a):
Which term is the dominant term? (Or you can use l'Hôpital's Rule.)

$\lim\limits _ { x \rightarrow 0 } \frac { \operatorname { sin } ^ { - 1 } ( x ) } { x ^ { 2 } }$
Hint (b):
Use l'Hôpital's Rule.

$\lim\limits _ { x \rightarrow 3 } \frac { x ^ { n } - 3 ^ { n } } { x - 3 }$
Hint (c):
$n$ is a number. The variable is $x$ .
Step (c):
$= \lim\limits_{x\to 3}nx^{n-1}$

$\lim\limits _ { x \rightarrow 1 ^ { - } } \frac { \operatorname { ln } ( 1 - x ) } { \operatorname { cot } ( \pi x ) }$
Hint (d):
Use l'Hôpital's Rule twice.

$\lim\limits _ { x \rightarrow \infty } \frac { x ^ { 2 } } { 2 ^ { x } }$
Step (e):
$=\lim\limits_{x\to\infty}\frac{2x}{2^x\ln(2)}$

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