Consider the function f(x)= cases x x , & x<0

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

Problem

Consider the function f(x)={sin⁡xx,x<0,k,x=0,x2−1x−1,0<x<2,4−x,x≥2.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][-1,3] in view. (a) Part 1: (i) Determine the value of kk that makes ff continuous at x=0x=0. (b) Part 2: (ii) Examine the continuity of ff in each of the sub-intervals (−1,0)(-1,0), (0,2)(0,2) and (2,3)(2,3), showing all necessary limit calculations. (c) Part 3: (iii) List every point in [−1,3][-1,3] where ff 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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