Vertical Asymptote

All functions need not have asymptotes. Most of the rational functions, exponential functions and hyperbolic functions have asymptotes.

If the asymptote is linear and parallel to y - axis then it is called vertical asymptote. A rational function p(x)/q(x) may have a vertical asymptote at x=a, for any ‘a’ where q (a) is 0.

If the asymptote is linear and parallel to x - axis then it is called as horizontal asymptote. Horizontal asymptotes occur in rational functions when one of the following conditions is met.

When the degree of the numerator is less than the degree of the denominator, then y=0 is the horizontal asymptote.
When the degree of the numerator is equal to the degree of the denominator then y=a/b where a is leading coefficient of the numerator and b is the leading coefficient of the denominator, is a horizontal asymptote.
When the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. The linear asymptote which is neither horizontal nor vertical is called slant or oblique asymptote.

When the numerator is of exactly one degree more than the denominator slant asymptote occurs. Using long division method, divide the numerator by the denominator and equate the quotient so obtained to y.

Whenever there is a common factor in the numerator and the denominator of a rational function we should cancel the common factor before finding the vertical asymptote. At the point obtained by equating this common factor to zero you will have a hole in the graph of the function.

Simple Asymptote Example
Question : Find all possible asymptotes and holes if any.

x2 + 2x - 151. f(x) = -------------- x2 + 7x + 10

(x+5)(x-3)
= ------------
(x+5)(x+2)
Vertical asymptotes; x = - 2
Horizontal asymptotes; y = 1/1 = 1
Slant asymptotes is none
Hole ; x = - 5



x2 - 5x + 82. g(x) = ------------- x - 3
x - 2
---------------
x - 3) x2 - 5x + 8
-x2 - 3x
----------------
-2x + 8
-2x + 6
----------------
- 2

= x - 2 . - 2
x - 3

Hence vertical asymptote is x = 3

Horizontal asymptote and hole is none

slant asymptote is y = x - 2