CPM Homework Help : APCALC Problem 10-30

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10-30.

Evaluate each limit below.

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

$\lim\limits _ { x \rightarrow 0 } \frac { \operatorname { tan } ( a x ) } { x }$
Hint (b):
Use l'Hôpital's Rule.
Think of $a$ as a number.

$\lim\limits _ { x \rightarrow \infty } ( x - \sqrt { x ^ { 2 } - 1 } )$
Hint (c):
Make the expression a fraction over $1$ , then rationalize the numerator.
The resulting expression can be used to evaluate the limit.

$\lim\limits _ { x \rightarrow 0 } \frac { 1 } { x } - \operatorname { csc } ( x )$
Step 1(d):
$=\lim \limits_{x\to 0}\Big(\frac{1}{x}-\frac{1}{\sin(x)}\Big)$
Step 2(d):
$=\lim \limits_{x\to 0}\frac{\sin(x)-x}{x\sin(x)}$
Step 3(d):
Use two iterations of l'Hôpital's Rule.

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