### Home > APCALC > Chapter 10 > Lesson 10.1.3 > Problem10-30

10-30.

Evaluate each limit below. Homework Help ✎

1. $\lim\limits _ { x \rightarrow 0 } \frac { \operatorname { sin } ^ { - 1 } ( x ) } { x }$

Use l'Hôpital's Rule.

1. $\lim\limits _ { x \rightarrow 0 } \frac { \operatorname { tan } ( a x ) } { x }$

Use l'Hôpital's Rule.
Think of $a$ as a number.

1. $\lim\limits _ { x \rightarrow \infty } ( x - \sqrt { x ^ { 2 } - 1 } )$

Make the expression a fraction over $1$, then rationalize the numerator.
The resulting expression can be used to evaluate the limit.

1. $\lim\limits _ { x \rightarrow 0 } \frac { 1 } { x } - \operatorname { csc } ( x )$

$=\lim_{x\to 0}\Big(\frac{1}{x}-\frac{1}{\sin(x)}\Big)$

$=\lim_{x\to 0}\frac{\sin(x)-x}{x\sin(x)}$

Use two iterations of l'Hôpital's Rule.