Question

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

Answer 1

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=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(x)` will have vertical asymptotes at those values of `x` for which the denominator is equal to zero.

2. The graph of `y = f(x)` will have horizontal asymptote if:
   a. `m > n` (the degree denominator `gt` numerator) then
   `y = f(x)` 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(x)` 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`