Prove that limit does not exist

[Deathfish]

Homework Statement

Prove that lim x->0 x(1+\frac{1}{x^2})^\frac{1}{2} does not exist.

The Attempt at a Solution

x² is always positive therefore \frac{1}{x^2} is always positive and (1+\frac{1}{x^2})^\frac{1}{2} is always positive.

therefore lim x->0+ = +x(1+\frac{1}{x^2})^\frac{1}{2}

however lim x->0- = -x(1+\frac{1}{x^2})^\frac{1}{2}

lim x->0+ \neq lim x-> 0-
lim x-> 0 does not exist.

I was thinking since this question is about proving if the limit exists or not so there is no need to evaluate it. Is there any better solution?

 

[Deathfish]

how do i use this thing

 

[lepton5]

Well, actually you are evaluating the existence of limit by continuity test. maybe if you want to more detail, you can make table for 0^+ (like 1/5, 1/10, 1/100, ...) vs result of limit and for 0^- (like -1/5, -1/10, ...).

and check it for left and right - limit (or you can make graph from the table), then you can conclude if that limit is exist or not.