What are the rules for finding oblique or slant asymptotes of rational functions?
Find oblique asymptote of R(x) = 3x/(x+4)
If there are none, please explain why.
consider the function f(x)
if lim([f(x)]/x as x tends to (+),(-) infinity is a and lim[f(x)-ax]= b as x tends to (+),(-) infinity then the line y=ax+b is a slant asymptote of f(x).
use this for any function f(x) to find oblique asymptotes.
yes you can do also long division but this applies only to rational functions what I gave you is for all functions...in your case the given function R(x) represents a Hyperbola whose asymptotes x=-4 and y=3 are parallel to the coordinate axis xx and yy therefore there are not oblique asymptotes.
As long as the numerator has degree n more than the denominator, $\displaystyle \frac{N(x)}{D(x)}= a_nx^n+ ...+ a_1x+ a_0+ \frac{R(x)}{D(x)}$.
In that case the "asymptote" is the polynomial part of that.
If the numerator has degree lower than the denominator then the polynomial is "0" and the asymptote is the horizontal line y= 0.