Question

How do you find the vertical, horizontal or slant asymptotes for `(x-5)/(x+1)`?

Answer 1

vertical asymptote x = - 1
horizontal asymptote y = 1

Explanation:

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

solve : x + 1 = 0 → x = -1 is the asymptote

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

divide all terms on numerator / denominator by x

`(x-5)/(x+1) = (x/x -5/x)/(x/x + 1/x ) = (1 - 5/x)/(1 + 1/x)`

now as x → ∞ , `5/x" and " 1/x → 0`

`rArr y = 1 " is the asymptote "`

Here is the graph of the function
graph{(x-5)/(x+1) [-20, 20, -10, 10]}