Mathematics

Problem

For a real parameter

k

, consider the family of functions

f_k(x)=\frac{kx^{4}+3x^{2}-2}{2x^{4}-k}.

(a) Derive an explicit formula for

L(k)=\displaystyle\lim_{x\to\infty}f_k(x)

in terms of

k

. Your derivation should clearly display the intermediate algebraic steps. (b) Determine all real values of

k

for which

L(k)

is positive, all values for which

L(k)

is negative, and any value(s) of

k

for which the limit does not exist. Support each conclusion with appropriate reasoning. (c) Identify every value of

k

for which

L(k)=\dfrac12

, and verify your result algebraically. a. Part (a) — Formula for

L(k)

b. Part (b) — Sign analysis and existence of limit c. Part (c) — Parameter values producing

L(k)=\tfrac12