How do you find the vertical, horizontal or slant asymptotes for y = (4 x + 6)/(x - 1)?

Dec 10, 2016

We have a vertical asymptote at $x = 1$ and horizontal asymptote at $y = 4$.

Explanation:

As $x - 1 \to 0$ i.e. $x \to 1$, $y \to \infty$ and hence, we have a vertical asymptote given by $x = 1$.

Further, we have $y = \frac{4 x + 6}{x - 1}$ and dividing numerator and denominator on right hand side by $x$, we get

$y = \frac{4 + \frac{6}{x}}{1 - \frac{1}{x}}$

and hence, as $x \to \infty$, $y \to \frac{4}{1} = 4$ i.e. we have a horizontal asymptote $y = 4$.

graph{(4x+6)/(x-1)[-20,20,-10,10]}