Consider the function defined as g(x)= cases x x , & x<0

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

Problem

Consider the function

g

defined as

g(x)=\begin{cases} \dfrac{\sin x}{x}, & x<0,\\ a+bx, & x\ge 0, \end{cases}

where

a

and

b

are real constants. (i) Find the values of

a

and

b

that make

g

continuous at

x=0

. (ii) For the obtained values of

a

and

b

, discuss the continuity of

g

on the entire real line. (iii) State with reasons the nature of discontinuity, if any, at

x=0

when

a

and

b

do not satisfy the condition found in part (i).