Question

How do you find the asymptotes for `f(x) = sqrt(x^2+5) - x`?

Answer 1

Horizontal asymptote : `y = 0 larr` See illustrative graph.

Explanation:

`f = sqrt(x^2+5)-x =( (sqrt(x^2+5)-x)( sqrt(x^2+5)+x))/ (sqrt(x^2+5)+x)`

`=(x^2+5-x^2)/ (sqrt(x^2+5)-x)+5/( sqrt(x^2+5)+x)`

`to 0`, as `x to oo`

So, y = 0 is the asymptote, .

graph{(sqrt(x^2+5)-x-y)y=0 [-10, 10, -5, 5]}