The piecewise function q(x)= cases x^ 2 -a^ 2 x-a , & x a, , & x=a cases contains the…

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

Problem

The piecewise function

q(x)=\begin{cases}\dfrac{x^{2}-a^{2}}{x-a}, & x\neq a,\\k, & x=a\end{cases}

contains the real constants

a

and

k

. (a) Explain in complete sentences what kind of discontinuity

q

can exhibit at

x=a

depending on the value of

k

. Support your explanation with the definition of continuity. (b) Derive the limit of

q(x)

x

approaches

a

in terms of

a

. (c) Determine the value of

k

that makes

q

continuous at

x=a

, and justify why no other value will suffice. (d) Suppose

a=5

and the function is redefined with the

k

you found in part (c). Describe verbally how the graph near

x=5

differs from the graph when

k

is any other real number.