Mathematics

Problem

The function

f

is defined by

\boxed{f(x)=\frac{3x^{2}-2x+1}{x^{2}+5}}.

Throughout this problem you will analyze the end behavior of

f

and its relationship to any asymptotes of the graph of

y=f(x)

. a. Part 1: (a) Evaluate

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

and

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

. b. Part 2: (b) Using your results from part (a), state the equations of all horizontal asymptotes of the graph of

y=f(x)

. c. Part 3: (c) Show algebraically whether the graph of

y=f(x)

intersects its horizontal asymptote(s). If an intersection exists, give the

x

–coordinate(s) of the point(s) of intersection and justify your conclusion.