Mathematics

Problem

For a real number

k

, define the piecewise function

p(x)=\begin{cases} \dfrac{x^2-4}{x-2}, & x<2,\\ kx+1, & x\ge 2. \end{cases}

Assume

p

is not otherwise restricted at

x=2

. Answer the following. a. Part 1: (a) Determine the value of

k

that guarantees the existence of

\displaystyle \lim_{x\to2}p(x)

. Show all algebraic steps leading to your result. b. Part 2: (b) For the value of

k

found in part (a), evaluate

\displaystyle \lim_{x\to2}p(x)

. c. Part 3: (c) Suppose

p

is now redefined so that

p(2)=7

. Discuss whether

p

is continuous at

x=2

under this new definition, citing the formal criteria for continuity in your explanation.