Mathematics

Problem

Consider the function

f(x)=\begin{cases} \dfrac{\sin x}{x}, & x<0,\\ k, & x=0,\\ \dfrac{x^{2}-1}{x-1}, & 0<x<2,\\ \sqrt{4-x}, & x\ge 2. \end{cases}

Answer the following questions keeping the closed interval

[-1,3]

in view. a. Part 1: (i) Determine the value of

k

that makes

f

continuous at

x=0

. b. Part 2: (ii) Examine the continuity of

f

in each of the sub-intervals

(-1,0)

(0,2)

and

(2,3)

, showing all necessary limit calculations. c. Part 3: (iii) List every point in

[-1,3]

where

f

is not continuous. For each point, state clearly whether the discontinuity is removable, jump or infinite, giving reasons based on the one-sided limits and the function value.

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