I do not quite understand this difference in limits

Question

According it my study material:

$\lim_{x\to 0^-}\frac {x}{|x|}= -1$ and $\lim_{x\to0^-} \frac {1}{|x|}= \infty$

Why does $\lim_{x\to0^-} \frac {1}{|x|}\ne -\infty$ as 1 still devided by a negative number?

Answer 1

If $x<0$, then $|x|=-x>0$. Therefore $$\lim_{x \to 0^-} \frac{1}{|x|} = \lim_{x \to 0^-} \frac{1}{-x} = - \lim_{x \to 0^-} \frac{1}{x} = -(-\infty)=+\infty.$$

Answer 2

The absolute value of a negative number is a positive number, i.e. $$ |x| = \left\{ \begin{array}{l l} x, & x > 0\\ -x, & x < 0 \end{array} \right. $$ so this means that $$ \lim_{x \rightarrow 0^- } \frac{1}{|x|} = \frac{1}{-x} =\infty, \ x < 0 $$

Answer 3

Remember that despite $x$ being negative, $|x|$ is still defined to be a strictly nonnegative number. Therefore you are actually dividing a positive number by a positive number, giving $\infty$ rather than $-\infty$.