A function is defined as follows: f(x)= cases (x+3) x x , & x0

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

A function

f

is defined as follows:

f(x)=\begin{cases} \displaystyle \frac{(x+3)\sin x}{x}, & x<0,\\[4pt] c, & x=0,\\[4pt] \displaystyle \frac{1-\cos(2x)}{k x^{2}}, & x>0. \end{cases}

Assume

c

and

k

are real constants. (a) Determine

\displaystyle\lim_{x\to 0^-}f(x)

and show all supporting work. (b) Express

\displaystyle\lim_{x\to 0^+}f(x)

in terms of

k

and justify each algebraic step. (c) Find any value of

k

for which the two one-sided limits found in parts (a) and (b) are equal. For that value of

k

, determine

c

so that

f

is continuous at

x=0

. (d) Using your conclusions from parts (a)–(c), classify the type of discontinuity (if any) that occurs at

x=0

for arbitrary real

k

and

c

. Provide a complete justification.