Question

Given f(x)=e^-x^3 find the vertical and horizontal asymptotes

Answer 1

Given:

`f\mleft(x\mright)=e^(-x^3)`

To find the vertical and horizontal asymptotes:

The line x=L is a vertical asymptote of the function f(x) if the limit of the function at this point is infinite.

But, here there is no such point.

Thus, the function f(x) doesn't have a vertical asymptote.

The line y=L is a vertical asymptote of the function f(x) if the limit of the function (either left or right side) at this point is finite.

`\begin{gathered} y=\lim _(x\rightarrow\infty)e^(-x^3) \\ =e^(-\infty) \\ y=0 \\ y=\lim _(x\rightarrow-\infty)e^(-x^3) \\ y=e^(\infty) \\ =\infty \end{gathered}`

Thus, y = 0 is the horizontal asymptote for the given function.