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

Question

2656 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-11-05T00:44:54

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

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 $(-1)/0^+ = -infty$ ,

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