# How do you find vertical, horizontal and oblique asymptotes for f(x)= (2x+3)/(3x+4)?

Mar 16, 2016

Vertical asymptote $x = - \frac{4}{3}$
Horizontal asymptote $y = \frac{2}{3}$

#### Explanation:

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation let the denominator equal zero.

solve : 3x + 4 = 0 → 3x = -4 → x = - 4/3" is the asymptote "

Horizontal asymptotes occur as lim_(x→±∞) f(x) → 0

divide all terms on numerator/ denominator by x

$\frac{2 x + 3}{3 x + 4} = \frac{\frac{2 x}{x} + \frac{3}{x}}{\frac{3 x}{x} + \frac{4}{x}} = \frac{2 + \frac{3}{x}}{3 + \frac{4}{x}}$

now as x →∞ , 3/x" and " 4/x → 0

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

Oblique asymptotes occur when the degree of the numerator is greater than the degree of the denominator. This is not the case here , hence there is no oblique asymptote.

Here is the graph of the function.
graph{(2x+3)/(3x+4) [-10, 10, -5, 5]}