1. help with finding limit

Could someone show me how to do these limits? Let p(x) be a polynomial of degree n: p(x) = a0 + a1x + a2x^2 + · · · + anx^n (an cannot equal 0). How would you prove that the limit from x to infinity of p(x)/anx^n is 1 and that the limit from x to negative infinity of p(x)/anx^n is 1.

Thank you for any help.

2. Originally Posted by MKLyon
Could someone show me how to do these limits? Let p(x) be a polynomial of degree n: p(x) = a0 + a1x + a2x^2 + · · · + anx^n (an cannot equal 0). How would you prove that the limit from x to infinity of p(x)/anx^n is 1 and that the limit from x to negative infinity of p(x)/anx^n is 1.

Thank you for any help.
For the first one:
$\lim_{x \to \infty} \frac{p(x)}{a_n x^n}$

$= \lim_{x \to \infty} \frac{a_n x^n + a_{n - 1}x^{n - 1} + ~ ... ~ + a_1x + a_0}{a_n x^n}$

Divide the numerator and denominator by $x^n$:
$= \lim_{x \to \infty} \frac{a_n + a_{n - 1}x^{-1} + ~ ... ~ + a_1x^{-n + 1} + a_0x^{-n}}{a_n}$

$= \lim_{x \to \infty} \left ( \frac{a_n}{a_n} + \frac{a_{n - 1}}{a_n} \frac{1}{x} + ~ ... ~ + \frac{a_1}{a_n} \frac{1}{x^{n - 1}} + \frac{a_0}{a_n} \frac{1}{x^n} \right )$

$= 1$

You do the second one.

-Dan