Question

What is `lim_ (xrarr0) (1/x -1/x^2-x)`?

Answer 1

Since both limits from left and right are `-infty`, the limit is also `-infty`.

Explanation:

Since `x->0`, the last term `-x` is irrelevant. So, let's focus on the rest.

We have that, if `x->0^-`, then `1/x -> -infty`, and so does `-1/x^2`. So we have `-infty-infty = -infty`.

On the other hand, if `x->0^+`, then we are in a `infty-infty` form, which we can't solve directly.

So, we must sum the two fractions:

`1/x-1/x^2 = (x^2-x)/x^3 = (x-1)/x^2`

And now the limit as `x->0^+` of this quantity is `(-1)/0^+ = -infty`,