### Home > CALC > Chapter 5 > Lesson 5.5.2 > Problem5-167

5-167.

Determine the following limits. For each, describe your method.

1. $\lim\limits_ { x \rightarrow 1 } \frac { \operatorname { ln } x } { x ^ { 2 } + 1 }$

The denominator does not equal $0$. Evaluate.

1. $\lim\limits_ { x \rightarrow 2 } \frac { x ^ { 3 } - 8 } { x ^ { 2 } - 4 }$

$x^3 −8 = (x − 2)(x^2 + 2x + 4)$

1. $\lim\limits_ { x \rightarrow \infty } \frac { \operatorname { ln } x } { \sqrt { x } }$

Think about the graphs of $y=\text{ln}x$ and $y=\sqrt{x}$. Which grows faster?

Is there a horizontal asymptote?

1. $\lim\limits_ { x \rightarrow 0 } \frac { \operatorname { sin } ^ { 2 } x } { \operatorname { sin } ( x ^ { 2 } ) }$

Use two iterations of l'Hôpital's Rule.

$1$