Say I have a limit of a fraction where x->infinity, which should I divide the equation with? The numerator's highest degree or the denominator's highest degree?
optional?
that's not what I said. once again, with a little more clarity ...
1. if the degree of the numerator = degree of the denominator, then divide every term by the highest degree variable.
2. if degree of the numerator > degree of the denominator, then the limit does not exist ... the rational function changes w/o bound.
3. if the degree of the numerator < degree of the denominator, then the limit is 0.
I believe that covers all three possibilities.
divide every term by $\displaystyle x^3$ ...
$\displaystyle \lim_{x \to \infty} \frac{1 + \frac{1}{x} - \frac{1}{x^3}}{\frac{1}{x} + \frac{1}{x^3}}
$
as x gets very large, the numerator approaches 1 and the denominator goes to 0 ...
in essence, it is the fraction $\displaystyle \frac{1}{a \, very \, small \, number}$
the value of the rational function increases w/o bound.