Let the function be defined by p(x)= cases r(x)=x^2-4x+6, & x<4

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

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. (1) 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. (2) 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). (3) 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. (4) 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.