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

Aug 5, 2016

vertical asymptote $x = \frac{3}{2}$
horizontal asymptote $y = \frac{3}{2}$

Explanation:

The denominator of f(x) cannot be zero as this is 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: 2x- 3 = 0 $\Rightarrow x = \frac{3}{2} \text{ is the asymptote}$

Horizontal asymptotes occur as

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

divide terms on numerator/denominator by x

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

as $x \to \pm \infty , f \left(x\right) \to \frac{3 + 0}{2 - 0}$

$\Rightarrow y = \frac{3}{2} \text{ is the asymptote}$

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)/(2x-3) [-20, 20, -10, 10]}