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5-167.

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

$\lim\limits_ { x \rightarrow 1 } \frac { \operatorname { ln } x } { x ^ { 2 } + 1 }$
Hint (a):
The denominator does not equal $0$ . Evaluate.

$\lim\limits_ { x \rightarrow 2 } \frac { x ^ { 3 } - 8 } { x ^ { 2 } - 4 }$
Hint (b):
$x^3 −8 = (x − 2)(x^2 + 2x + 4)$

$\lim\limits_ { x \rightarrow \infty } \frac { \operatorname { ln } x } { \sqrt { x } }$
Hint 1(c):
Think about the graphs of $y=\text{ln}x$ and $y=\sqrt{x}$ . Which grows faster?
Hint 2(c):
Is there a horizontal asymptote?

$\lim\limits_ { x \rightarrow 0 } \frac { \operatorname { sin } ^ { 2 } x } { \operatorname { sin } ( x ^ { 2 } ) }$
Hint (d):
Use two iterations of l'Hôpital's Rule.
Answer (d):
$1$

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