Mathematics

Problem

A rational function

g

is defined by

g(x)=\frac{7x^{4}-2x^{2}+9}{-3x^{4}+5x^{3}+1}.

(a) Determine

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

. Show all algebraic steps that support your result. (b) Determine

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

. Clearly indicate each stage of your work. (c) State the equation(s) of any horizontal asymptote(s) of

g

, justifying each with reference to your limits from parts (a) and (b). (d) Use the limits you have calculated to describe the end behavior of the graph of

g

, explicitly comparing its behavior for large negative values of

x

with that for large positive values of

x

. a. Part (a) — Limit as

x\to-\infty

b. Part (b) — Limit as

x\to\infty

c. Part (c) — Horizontal asymptote(s) d. Part (d) — Comparison of end behavior

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