Single Variable Calculus, Chapter 2, Review Exercises, Section Review Exercises, Problem 1

Given the graph of $f$







a.) Determine each limit if it exist. If the limit does not exist. Explain why.

$\qquad$ i.) $\displaystyle \lim \limits_{x \to 2^+} f(x)$

$\qquad \qquad $ Referring to the given graph. $\displaystyle \lim \limits_{x \to 2^+} f(x) = 3$

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

$\qquad \qquad $ Referring to the given graph $\displaystyle \lim \limits_{x \to -3^+} f(x) = 0$

$\qquad$ iii.) $\displaystyle \lim \limits_{x \to -3} f(x)$

$\qquad \qquad $ Referring to the given graph $\displaystyle \lim \limits_{x \to -3} f(x)$ does not exist because left and right hand limits approaches different values.

$\qquad$ iv.) $\displaystyle \lim \limits_{x \to 4} f(x)$

$\qquad \qquad $ Referring to the given graph $\displaystyle \lim \limits_{x \to 4} f(x) = 2$

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

$\qquad \qquad $ Referring to the given graph $\displaystyle \lim \limits_{x \to 0} f(x) = \infty$

$\qquad$ vi.) $\displaystyle \lim \limits_{x \to 2^-} f(x)$

$\qquad \qquad $ Referring to the given graph $\displaystyle \lim \limits_{x \to 2^-} f(x) = -\infty$

b.) Indicate the equations of the vertical asymptotes.

$\qquad$ The vertical asymptotes happens when the value of $x$ is not defined. Referring to the graph, the vertical asymptotes are $x = 0, x = 2$

c.) Where is $f$ discontinuous? Explain.

$\qquad$ $f$ is discontinuous at $x = -3$ because of jump discontinuity. Also, the function is discontinuous at $x = 0$ and $x = 2$ because of infinite discontinuity. Lastly, $f$ is discontinuous at $x = 4$ because of removable discontinuity.