For each real parameter

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

Problem

a

, define the function

f_a(x)=\frac{e^{ax}+x}{e^{x}}, \quad x\in\mathbb R.

Investigate how the horizontal asymptotes of

f_a

depend on the value of

a

. (a) For fixed

a

, evaluate

\displaystyle \lim_{x\to\infty} f_a(x)

in terms of

a

. Provide a complete derivation. (b) Determine all values of

a

for which the limit found in part (a) is finite. Justify your answer thoroughly. (c) For those values of

a

that yield a finite limit, write the corresponding horizontal asymptote equation. (d) Evaluate

\displaystyle \lim_{x\to -\infty} f_a(x)

when

a>1

a=1

, and

a<1

, and discuss whether additional horizontal asymptotes arise as

x\to -\infty

. Provide rigorous support for each case.