# How do you find the vertical, horizontal or slant asymptotes for (3x + 5) /( x - 2)?

Aug 13, 2016

vertical asymptote at x = 2
horizontal asymptote at y = 3

#### Explanation:

The denominator of the function cannot be zero as this would make it undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

solve: x - 2 = 0 $\Rightarrow x = 2 \text{ is the asymptote}$

Horizontal asymptotes occur as

${\lim}_{x \to \pm \infty} , y \to c \text{ (a constant)}$

divide terms on numerator/denominator by x.

$\frac{\frac{3 x}{x} + \frac{5}{x}}{\frac{x}{x} - \frac{2}{x}} = \frac{3 + \frac{5}{x}}{1 - \frac{2}{x}}$

as $x \to \pm \infty , y \to \frac{3 + 0}{1 - 0}$

$\Rightarrow y = 3 \text{ is the asymptote}$

Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here ( both of degree 1 ) Hence there are no slant asymptotes.
graph{(3x+5)/(x-2) [-15.8, 15.8, -7.9, 7.9]}