A fx frac x 1 9 x 2 4x 21 b g x frac 2x sqrt x 2 find all vertical asymptotes x a. If the degrees of the numerator and denominator are equal take the coefficient of the highest power of x in the numerator and divide it by the coefficient of the highest power of x in the denominator.

Calculus I Limits Finding Limits Algebraically Piecewise Functions 1 Calculus Youtube Topics

### Find the equations of the horizontal and vertical asymptotes.

**How to find horizontal asymptotes with limits**. 1 put equation or function in y form. Whether or not a rational function in the form of r x p x q x has a horizontal asymptote depends on the degree of the numerator and denominator polynomials p x and q x. Finding a limit of a rational function.

Rules for finding horizontal asymptotes now that we have a grasp on the concept of degrees of a polynomial we can move on to the rules for finding horizontal asymptotes. That quotient gives you the answer to the limit problem and the heightof the asymptote. X 1 0 x 1 thus the graph will have a vertical asymptote at x 1.

Find the vertical and horizontal asymptotes of the graph of f x x2 2x 2 x 1. F x displaystyle frac 1 frac1 x 1 frac 6 x frac 5 x 2 then take the limit it should be y to1 x to pm infty. Confirm analytically that y 1 is the horizontal asymptote of f x frac x 2 x 2 4 as approximated in example 29.

Before using theorem 11 let s use the technique of evaluating limits at infinity of rational functions that led to that theorem. 2 multiply out expand any factored polynomials in the numerator or denominator. The vertical asymptotes will occur at those values of x for which the denominator is equal to zero.

To find horizontal asymptotes. 3 remove everything except the terms with the biggest exponents of x found in the numerator and denominator. For your horizontal asymptote divide the top and bottom of the fraction by x 2.

To nd the horizontal asymptote we note that the degree of the numerator.