CPM Homework Help : PC3 Problem 8-105

Let $f ( x ) = \left\{ \begin{array} { l l } { 2 ^ { x } } & { \text { for } x < 2 } \\ { 4 } & { \text { for } x \geq 2 } \end{array} \right.$ . Evaluate:

$\lim\limits_{ x \rightarrow 2 } f ( x )$
Hint (a):
The limit must approach the same height from both sides in order to exist.

$\lim\limits_{ x \rightarrow 5 } f ( x )$
Hint (b):
Which piece of the function should you use?

Is the function continuous at $x = 2$ ? Use the formal definition of continuity to justify your answer.
Hint (c):
For a function to be continuous at a point, the one-sided limits must agree and they must equal the value of the function at that point.
Is the function continuous at $x = 5$ ? Use the formal definition of continuity to justify your answer.