How do you find the asymptotes for #R(x)=(3x+5) /(x-6)#?

Vertical asymptotes occur when the denominator of the rational function is 0.

In this question this would occur when x - 6 = 0 ie x = 6

[ Horizontal asymptotes can be found when the degree of the

numerator and the degree of the denominator are equal. ]

Here they are both of degree 1 and so are equal.

The horizontal asymptote is found by taking the ratio of leading

coefficients .

To find the asymptotes of the rational function ( R(x) = \frac{3x + 5}{x - 6} ), we need to identify any vertical, horizontal, and slant asymptotes.

1. Vertical Asymptote: Vertical asymptotes occur where the denominator of the function becomes zero, but the numerator doesn't. In this case, the vertical asymptote occurs when ( x - 6 = 0 ). Solving for ( x ), we get ( x = 6 ). So, the vertical asymptote is the line ( x = 6 ).

2. Horizontal Asymptote: Horizontal asymptotes can be determined by comparing 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 ( 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 degree of the numerator (1) is less than the degree of the denominator (1), so the horizontal asymptote is ( y = 0 ).

3. Slant Asymptote (if applicable): Slant asymptotes occur when the degree of the numerator is one more than the degree of the denominator. In this case, since the degrees are equal, there is no slant asymptote.

To summarize:

• Vertical asymptote: ( x = 6 )
• Horizontal asymptote: ( y = 0 )
• No slant asymptote.