Mathematics

Problem

A family of rational functions is given by

F_k(x)=\frac{x^2-kx}{x-k}, \quad k\neq 0.

Throughout this problem,

k

is a real parameter. (a) Show that

\lim_{x\to k} F_k(x)

can be expressed as a linear function of

k

. State this linear function explicitly. (b) For which value(s) of

k

does

\lim_{x\to k} F_k(x)=4

? Provide complete algebraic justification. (c) Discuss whether there exists any real value of

k

for which the limit in part (a) fails to exist. Support your conclusion with appropriate reasoning.