Question

How do I find the asymptotes for a rational function? (Oblique, Horizontal, and Vertical)

Answer 1

If it's a rational function, you can find any horizontal or oblique (aka End Behavior) asymptotes by dividing the numerator by the denominator. To find a vertical asymptote, simply set the denominator of your rational function = 0 (after cancelling out any common factors) and find the x-values that make the denominator vanish. For example, if you were looking at \[(x ^{2}+5)/x\] division would yield an oblique asymptote at y=x (the remainder term, 5/x, would vanish for large x-values and be insignificant) and setting the denominator equal to zero would give a vertical asymptote at x=0.

Answer 2

THANKS

Answer 3

Suppose that we have a rational function of the form...... y=f(x)/g(x)=ax^p/bx^q, where p is the largest power in the numerator, q is the largest power in the denominator and a and b are the coefficients of each respectively. Then,

set the denominator equal to zero and solve ( find roots of expression in denominator) the zeroes (if any) are the vertical asymptotes

if p < q, there will be a horizontal asymptote at y = 0.
if p = q, there will be a horizontal asymptote at y=a/b.
if p = q + 1, there will be an oblique (slant) asymptote.
if p > q + 1, there will not be a horizontal or oblique asymptote