Mathematics

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. a. (i) Find the values of

a

and

b

that make

g

continuous at

x=0

. b. (ii) For the obtained values of

a

and

b

, discuss the continuity of

g

on the entire real line. c. (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).