Define the function q(x)= cases x^2-1 x-1 , & x<1, [6pt] 4-x^2 , & 1 x<2, [6pt]…

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

Problem

Define the function

q(x)=\begin{cases} \dfrac{x^2-1}{x-1}, & x<1,\\[6pt] \sqrt{4-x^2}, & 1\le x<2,\\[6pt] \ln(x-1), & x\ge 2. \end{cases}

Respond to each part below. Unless otherwise noted, all limits are taken using exact values (no decimal approximations). (a) Compute

\displaystyle \lim_{x\to1^-}q(x)

. (b) Compute

\displaystyle \lim_{x\to1^+}q(x)

. (c) Does

\displaystyle \lim_{x\to1}q(x)

exist? Justify your answer with reference to parts (a) and (b). (d) Compute

\displaystyle \lim_{x\to2^-}q(x)

. (e) Compute

\displaystyle \lim_{x\to2^+}q(x)

, or explain why the expression is not defined. (f) Based on all preceding work, list every value of

c

in the interval

(-\infty,3]

at which

q

fails to possess a two-sided limit. Provide a brief justification for each listed value.