Evaluate each of the following limits. For each part, describe your method.

1. $\lim\limits _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } + 1 }$ 

   Hint (a):

   The denominator does not equal $0$. Evaluate.

2. $\lim\limits _ { x \rightarrow 2 } \frac { x ^ { 3 } - 8 } { x ^ { 2 } - 4 }$ 

   Hint (b):

   $x^{3} −8 = \left(x − 2\right)\left(x^{2} + 2x + 4\right)$

3. $\lim\limits _ { x \rightarrow \infty } \frac { \sqrt { x } } { 3 x + 4 }$ 

   Hint (c):

   Since this is a limit to infinity, which function grows faster, the function in the numerator or the function in the denominator?

4. $\lim\limits _ { x \rightarrow 0 } \frac { \operatorname { sin } ^ { 2 } ( x ) } { \operatorname { sin } ( x ^ { 2 } ) }$ 

   Hint (d):

   Use two iterations of l'Hôpital's Rule.