How to find the asymptotes of #R(x)=(3x+5) / (x-6)#?

Question

Lincoln Anderson · Accepted Answer

Vertical: #x=6#
Horizontal: #y=3# Vertical asymptotes are values for which the function is undefined. Our function is undefined when the denominator is equal to zero. Setting it equal to zero, we get #x=6#. This is our vertical asymptote. To find our horizontal asymptote, we need to analyze the degrees of the function. We have the same degree on the top and bottom, so the variables cancel. As a horizontal asymptote, we're left with #y=3# Hope this helps!

Sadie Ackerman · Answer

undefined To find the asymptotes of $ R(x) = \frac{3x + 5}{x - 6} $, you need to consider the behavior of the function as $ x $ approaches certain values. Horizontal asymptotes occur when $ x $ approaches positive or negative infinity. Vertical asymptotes occur when the denominator of the function equals zero, but the numerator does not. In this case:

1. To find the vertical asymptote, set the denominator $ x - 6 $ equal to zero and solve for $ x $. $ x - 6 = 0 $ implies $ x = 6 $. Therefore, there is a vertical asymptote at $ x = 6 $.

2. To find the horizontal asymptote, compare the degrees of the numerator and the denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at $ y = 0 $. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients. In this case, the degree of the numerator (1) is less than the degree of the denominator (1). Therefore, the horizontal asymptote is at $ y = 0 $.

So, the vertical asymptote is $ x = 6 $, and the horizontal asymptote is $ y = 0 $.