# How do you find the vertical, horizontal or slant asymptotes for  y = 6/x?

Mar 26, 2016

we have a vertical asymptote at $x = 0$
we have a horizontal asymptote at $y = 0$
graph{6/x [-13.38, 16.53, -7.87, 7.09]}

#### Explanation:

Given: $y = \frac{6}{x}$
Required vertical, horizontal or slanted asymptotes?
Solution Strategy: Definition and principles governing asymptotes.

Asymptotes Rule:
Let f be the (reduced) rational function
f(x) = (a_nx^n + · · · + a_1x + a_0)/(b_mx^m + · · · + b_1x + b_0)

1. The graph of $y = f \left(x\right)$ will have vertical asymptotes at those values of $x$ for which the denominator is equal to zero.

2. The graph of $y = f \left(x\right)$ will have horizontal asymptote if:
a. $m > n$ (the degree denominator $>$ numerator) then
$y = f \left(x\right)$ will have a horizontal asymptote at y = 0 (x-axis)
b. If $m = n$ (degree of numerator and denominator are the same),
then $y = f \left(x\right)$ will have a horizontal asymptote at $y = {a}_{n} / {b}_{m}$
c. If $m < n$ (numerator degree is larger than denominator), then the graph of y = f(x) will have no horizontal asymptote

From 1) we have a vertical asymptote at $x = 0$
From 2a $m = 1 > n = 0$ thus we have a horizontal asymptote at $y = 0$