How do you find all the asymptotes for function #f(x) = (7x+1)/(2x-9)#?

Question

Elias Ackerman · Accepted Answer

First we find the vertical asymptote.
That is when the numerator gets near to #0#:
#2x-9~~0->x~~4 1/2# The horizontal is found when we make #x# larger and larger. The #+1# and #-9# then matter less and less, and the funtion begins to approach #f(x)~~(7x)/(2x)=7/2=3 1/2#
 graph{(7x+1)/(2x-9) [-65.8, 65.9, -32.95, 33]}

Serenity Johnson · Answer

undefined To find the asymptotes of the function $ f(x) = \frac{7x + 1}{2x - 9} $, we examine the behavior of the function as $ x $ approaches certain values.

1. Vertical Asymptote:
   Vertical asymptotes occur where the denominator of the function becomes zero but the numerator does not. Set the denominator equal to zero and solve for $ x $:
   $$ 2x - 9 = 0 $$
   $$ x = \frac{9}{2} $$
   Therefore, there is a vertical asymptote at $ x = \frac{9}{2} $.

2. Horizontal Asymptote:
   To find the horizontal asymptote, compare the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at $ y = 0 $. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is greater, there is no horizontal asymptote.
   In this case, the degrees of the numerator and denominator are both 1. Thus, the horizontal asymptote is the ratio of the leading coefficients:
   $$ y = \frac{7}{2} $$

Therefore, the asymptotes for the function $ f(x) = \frac{7x + 1}{2x - 9} $ are:
- Vertical asymptote: $ x = \frac{9}{2} $
- Horizontal asymptote: $ y = \frac{7}{2} $