How do you find the vertical, horizontal or slant asymptotes for f(x)= (4x+8)/(x-3)?

Mar 3, 2016

$V A : x - 3 = 0 \to x = 3 , H A : f \left(x\right) = \lim x \to \propto \frac{4 x}{x} = 4$

Explanation:

Take the denominator set it equal to 0 and solve for x to find the vertical asymptote. To find the horizontal asymptote find the limit as x goes to infinity and use the end behavior of a rational function. That is, pick the highest degree term from the top and bottom then simplify

$x = 3$ is a Vertical Asymptote
$y = 4$ is a Horizontal Asymptote
Slant Asymptote: None

Explanation:

At $x = 3$ the value of the function is not defined. Therefore the line approached by the graph but will never touch it no matter how far is $x = 3$

Similarly, no matter how far the graph of the function increases or decreases to the right or to the left respectively, the line approached by the graph is $y = 4$

Therefore $y = 4$ is a horizontal asymptote

See the graph of $y = \frac{4 x + 8}{x - 3}$

graph{(y- (4x+8)/(x-3))=0[-40,40,-20,20]}

See also the graph of the asymptotes $x = 3$ and $y = 4$

graph{(y-4)(y+(1000)x-3*(1000))=0[-40,40,-20,20]}

God bless ... I hope the explanation is useful