Mathematics

Problem

Consider the function

f

defined by

f(x) = \begin{cases} 4x^2 - 5x + 1, & x < 2, \\ kx + 7, & x \ge 2, \end{cases}

where

k

is a real constant. a. Part 1: (a) Compute

\displaystyle \lim_{x\to 2^-} f(x)

and

\displaystyle \lim_{x\to 2^+} f(x)

in terms of

k

. Show your work. b. Part 2: (b) Determine the value of

k

for which

f

is continuous at

x = 2

. Justify your answer algebraically. c. Part 3: (c) Using the value of

k

found in part (b), evaluate

\displaystyle \lim_{x\to 2} f(x)