Mathematics

Mathematics

Problem

A function qq is described as follows: • For x≤−1x\le -1, q(x)=2q(x)=2 (a horizontal line). • For −1<x<1-1<x<1, q(x)=1−x2q(x)=\sqrt{1-x^{2}} (the upper half of a circle of radius 11 centered at the origin). • For x≥1x\ge 1, q(x)=x−1q(x)=x-1 (a line of slope 11). Which of the following statements is true? a) lim⁡x→−1q(x)=0, and q is continuous at x=1.\displaystyle \lim_{x\to -1} q(x)=0,\text{ and } q \text{ is continuous at } x=1. b) lim⁡x→−1q(x) does not exist, and q is continuous at x=1.\displaystyle \lim_{x\to -1} q(x)\text{ does not exist, and } q \text{ is continuous at } x=1. c) lim⁡x→−1q(x) does not exist, and q is discontinuous at x=1.\displaystyle \lim_{x\to -1} q(x)\text{ does not exist, and } q \text{ is discontinuous at } x=1. d) lim⁡x→−1q(x)=2, and q is discontinuous at x=1.\displaystyle \lim_{x\to -1} q(x)=2,\text{ and } q \text{ is discontinuous at } x=1.

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