1. ## Limits problem

I am supposed to find the limits of these two problems and I'm not sure how. Can someone explain these to me? Thanks

Lim $(1/x-1/abs(x))$
Left hand limit of 0

Lim $(1/x-1/abs(x))$
Right hand limit of 0

Sorry I'm not sure how to get the code thing to work. Those are supposed to be fractions. Thanks....

2. You have to remember that $|x|$ is defined as

$|x|=\begin{cases}\phantom{-}x\textrm{ if }x \geq 0\\-x\textrm{ if }x<0\end{cases}$.

So if you're approaching $0$ from the left (where $x < 0$) that would mean

$\frac{1}{x} - \frac{1}{|x|} = \frac{1}{x} - \frac{1}{-x}$

$= \frac{1}{x} + \frac{1}{x}$

$= \frac{2}{x}$.

Therefore $\lim_{x \to 0^-}\left(\frac{1}{x} - \frac{1}{|x|}\right) = \lim_{x \to 0^-}\left(\frac{2}{x}\right)$

$= -\infty$.

Now follow a similar procedure where you approach $0$ from the right.

3. So from the right, it would equal 0, right?

4. Correct.

5. The guide says it doesn't exist from the left. Why's that? Is it the same from the right?

6. Originally Posted by Godzilla
The guide says it doesn't exist from the left. Why's that? Is it the same from the right?
Technically an "infinite limit" doesn't exist because it doesn't tend to a finite value. By writing $-\infty$ it means that it decreases without bound.