Question

How do you determine the limit of `((x^2 +2)/(x^3-1))` as x approaches 0?

Answer 1

I found `-2`

Explanation:

I would try by approaching zero from the left and the right where you get, in both cases, that your function get as near as possible to the value `-2`.

Your function is also continuous in `x=0` giving you:

`lim_(x->0)f(x)=f(x_0)`

i.e., the limit at zero is equal to the function evaluated at zero or:
`f(0)=((0^2+2)/(0^3-1))=-2`

I would say that the limit exists and is equal to `-2`.

Graphically you can see this as well:

graph{(x^2+2)/(x^3-1) [-10, 10, -5, 5]}