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

Jul 30, 2016

vertical asymptote x = 2
horizontal asymptote y = 3

Explanation:

The denominator of f(x) cannot be zero as this would be 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 of x then it is a vertical asymptote.

solve : x - 2 = 0 → x = 2 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

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

as $x \to \pm \infty , f \left(x\right) \to \frac{3 + 0}{1 - 0}$

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

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) [-20, 20, -10, 10]}