The function is defined by h(x)= cases 1 x , & x0

MATHEMATICS · Single correct · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

The function

h

is defined by

h(x)=\begin{cases} \dfrac{1}{x}, & x<0,\\[4pt] \sin\!\left(\dfrac{1}{x}\right), & x>0. \end{cases}

On the window

-1\le x\le 1

-10\le y\le 10

, the left branch shoots off the screen as

x\to 0^{-}

, while the right branch appears nearly horizontal near

y=0

. Which statement is true about

\displaystyle\lim_{x\to 0}h(x)

? (a) The limit does not exist because

h(x)\to -\infty

x\to 0^{-}

and oscillates between

-1

and

1

x\to 0^{+}

. (b) The limit equals

0

because

\sin\left(\dfrac{1}{x}\right)\to0

x\to0

. (c) The limit equals

-\infty

since the unbounded left–hand branch dominates any right–hand behavior. (d) The limit does not exist simply because

h

is undefined at

x=0