For a real parameter

MATHEMATICS · Multi-part · AP · AP Precalculus · Polynomial and Rational Functions · Polynomial and Rational Functions

Problem

k

, define the piecewise function

f_k(x)=\begin{cases}\dfrac{x^{2}-4}{x-k}, & x\ne k,\\[6pt]c, & x=k,\end{cases}

where

c

is a real constant. (a) For general

k

, determine all real numbers at which

f_k

is not defined before the value of

c

is assigned. (b) Find all value(s) of

k

for which

f_k

possesses exactly one vertical asymptote on the real number line. Fully justify your reasoning, including any necessary limit evaluations. (c) For the value(s) of

k

obtained in part (b), compute the left–hand and right–hand limits of

f_k(x)

x

approaches the vertical asymptote, indicating the sign of divergence on each side. (d) For the same value(s) of

k

, determine every horizontal or oblique asymptote of

f_k

and justify your conclusion. (e) Decide whether a choice of

c

exists that would remove any discontinuity at

x=k

. If so, find all such

c

and explain how your result affects the overall continuity of

f_k

Related theory

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