Question

How do you find the limit `(x^3-6x-2)/(x^3+4)` as `x->2`?

Answer 1

In the end, you find it by substitution.

Explanation:

The reasoning may go as follows:

If `x` is very close to `2`, then `x^3` is close to `8`, so

`x^3+4` is close to `12`.

Also, if `x` is close to `2`, then `x^3-6x-2` is close to `8-12-2 = -6`

Therefore,

`(x^3-6x-2)/(x^3+4)` is close to `(-6)/12 = -1/2`.