Without a calculator, determine the following limits.

1. $\lim\limits _ { x \rightarrow \infty } [ 4 x + 2 - \frac { 3 } { x - 2 } ]$

   Hint (a):

   If you took the limit of each term separately, would any of those limits $= 0$?

2. $\lim\limits _ { x \rightarrow - \infty } \frac { 2 x ^ { 2 } + x - 21 } { 2 x ^ { 2 } + 5 x - 7 }$

   Hint (b):

   Check the end behavior of the graph by comparing the highest powers from both the numerator and the denominator.

   Partial Answer (b):

   $\lim\limits_{x\rightarrow -\infty }\frac{2x^{2}}{2x^{2}}=\lim\limits_{x\rightarrow -\infty }1=\text{ \_\_\_\_ }$

3. $\lim\limits _ { x \rightarrow 0 ^ { + } } [ \operatorname { ln } x + 18 ]$

4. $\lim\limits _ { x \rightarrow \frac { \pi } { 3 } } \frac { \operatorname { sin } x - \frac { \sqrt { 3 } } { 2 } } { x - \frac { \pi } { 3 } }$
   

   Hint (d):

   This is Ana's Definition of the Derivative;
   Find $f^\prime(a)$.
   $f'(a)=\lim\limits _{x\rightarrow a}\frac{f(x)-f(a)}{x-a}$