1. ## Infinity proving

Suppose f : D --> R is a function whose domain contains (-infinity, a) for some a$\displaystyle \in$R. We define $\displaystyle lim x--> - infinity f(x) = infinity$to mean for all M >0 there exists K <0 such that

$\displaystyle x < K ==> f(x) > M$.

We define $\displaystyle lim x--> - infinity f(x) = - infinity$ to mean for all M < 0 there exists K < 0 such that

$\displaystyle x < K ==> f(x) < M$.

Suppose n is a positive integer and let $\displaystyle P:R --> R$ be a polynomial function defined by
$\displaystyle P(x) = anx^n + an-1x^(n-1) + ... + a0$ for all x $\displaystyle \in$ R.

(a) Suppose that n is odd and an is positive. Prove that $\displaystyle lim x--> - infinity P(x) = - inifinity$. (Hint: You may use that $\displaystyle x^n < x$for all x < -1)

(b) Prove that if n is even and an is positive then $\displaystyle lim x--> - infinity P(x) = inifinity$. (Hint: replacing P with -P) Do not do this
$\displaystyle lim x--> - infinity -P(x)$ = $\displaystyle - lim x--> -infinity P(x)$ = $\displaystyle - (- infinity) = infinity$.

(c) Prove that if n is even and an is positive then $\displaystyle lim x--> - infinity P(x) = infinity$.