Question

How do you find the asymptotes for `g(x)=x/(root4(x^4+2))`?

Answer 1

Evaluate the limits of `g(x)` as `x->+oo` and `x->-oo` to find horizontal asymptotes `y=1` and `y=-1`.

Explanation:

`g(x) = x/root(4)(x^4+2) = x/abs(x) abs(x)/root(4)(x^4+2) = x/abs(x) 1/root(4)(1+2/x^4)`

So `lim_(x->+oo) g(x) = 1` and `lim_(x->-oo) g(x) = -1`

So `g(x)` has horizontal asymptotes `y=1` and `y=-1`

`x^4+2 >= 2 > 0` for any Real number `x`

Hence the denominator of `g(x)` is always non-zero and `g(x)` has no vertical asymptotes.

graph{x/root(4)(x^4+2) [-5.55, 5.55, -2.775, 2.774]}