Let , where , , and the dots denote lower-degree terms. Answer the following…

MATHEMATICS · Multi-part · AP · AP Precalculus · Polynomial and Rational Functions · Polynomial and Rational Functions

Problem

Let

s(x)=a x^{2m}+b x^{2m-1}+\dots+d

, where

a\neq0

m\in\mathbb{N}

, and the dots denote lower-degree terms. Answer the following questions: (i) Prove rigorously that

\displaystyle\lim_{x\to\pm\infty}s(x)=+\infty

when

a>0

and

\displaystyle\lim_{x\to\pm\infty}s(x)=-\infty

when

a<0

. (ii) For

t(x)=-2x^{8}+5x^{4}-1

, evaluate

t(1000)

and

t(-1000)

, and explain how these numerical values illustrate the result proved in part (i). (iii) Comment on how the even degree of

s(x)

influences the minimum possible number of

x

-axis crossings of its graph, giving a short justification based on end behaviour.