Mathematics

Problem

Let the function

p

be defined by

p(x)=\begin{cases} r(x)=x^2-4x+6, & x<4,\\ s(x)=\dfrac{(k-1)x^2+8x-k}{x-4}, & x\ge 4, \end{cases}

where

k

is a real parameter. a. Part 1: (a) Explain why the existence of

\displaystyle \lim_{x\to 4^-} p(x)

is immediate from the algebraic form of

r(x)

, and state that limit. b. Part 2: (b) Determine the value(s) of

k

for which

\displaystyle \lim_{x\to 4^+} p(x)

exists and equals the value obtained in part (a). c. Part 3: (c) Using the

k

value(s) from part (b), evaluate

\displaystyle \lim_{x\to 4} p(x)

and confirm that the two one–sided limits are equal. d. Part 4: (d) Suppose a student claims that multiplying the numerator and denominator of

s(x)

(x-4)

is a valid technique for finding the limit in part (b). Critically assess this claim and describe a correct alternative approach.