Single Variable Calculus, Chapter 2, Review Exercises, Section Review Exercises, Problem 23

Suppose that

$\qquad f(x) = \left\{ \begin{array}{ccc} \sqrt{-x} & \text{ if } & x < 0 \\ 3 - x & \text{ if } & 0 \leq x < 3 \\ (x - 3)^2 & \text{ if } & x > 3 \end{array} \right.$

a.) Find each limit. if it exists

$\qquad$ i.)$\lim \limits_{x \to 0^+} f(x)$

$\qquad \qquad$ Using Equation 2, $\lim \limits_{x \to 0^+} f(x) = \lim \limits_{x \to 0^+} (3 - x) = 3-0 = 3$

$\qquad$ ii.) $\lim \limits_{x \to 0^-} f(x)$

$\qquad \qquad$ Using Equation 1, $\lim \limits_{x \to 0^-} f(x) = \lim \limits_{x \to 0^-} \sqrt{-x} = \sqrt{0} = 0$

$\qquad$ iii.) $\lim \limits_{x \to 0} f(x)$

$\qquad \qquad$ Referring to the given conditions, $\lim \limits_{x \to 0} f(x)$ does not exist

$\qquad$ iv.) $\lim \limits_{x \to 3^-} f(x)$

$\qquad \qquad$ Using Equation 2, $\lim \limits_{x \to 3^-} f(x) = \lim \limits_{x \to 3^-} (3 -x) = 3 - 3 = 0$

$\qquad$ v.) $\lim \limits_{x \to 3^+} f(x)$

$\qquad \qquad$ Using Equation 3, $\lim \limits_{x \to 3^+} f(x) = \lim \limits_{x \to 3^+} (x - 3)^2 = (3 - 3)^2 = 0$

$\qquad$ vi.) $\lim \limits_{x \to 3} f(x)$

$\qquad \qquad$ Referring to the given conditions, $\lim \limits_{x \to 3} f(x) = 0$

b.) Find where $f$ is discontinuous.

$\qquad$ Referring to the given conditions $f$ is discontinuous at $x = 0$ because $\lim \limits_{x \to 0} f(x)$ does not exist and also $f$ is discontinuous at $x = 3$ because $f(3)$ is undefined.

$\qquad$ Therefore $f$ is discontinuous at 0 and 3.

c.) Graph the function $f$.