
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
$\displaystyle P(x) = anx^n + an1x^(n1) + ... + a0$ 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 $\displaystyle lim x>  infinity P(x) = infinity$.