# Chapter 1 - Functions and Limits - 1.6 Calculating Limits Using the Limit Laws - 1.6 Exercises: 45

$-\infty$, (the limit does not exist)

#### Work Step by Step

Since $|x|=-x$ for $x<0$, we have $|x|=\left\{\begin{array}{lll} x & if & x \geq 0\\ -x & if & x < 0 \end{array}\right.$ Approaching $x=0$ from the LEFT, means $x < 0$... $\displaystyle \lim_{x\rightarrow 0-}(\frac{1}{x}-\frac{1}{|x|})=\lim_{x\rightarrow 0-}(\frac{1}{x}-\frac{1}{-x})=\lim_{x\rightarrow 0-}\frac{2}{x}$, (see sec.1-5, infinite limits) $\displaystyle \frac{2}{x}$ is negative and increases without bound in magnitude, so the limit does not exist $(\displaystyle \lim_{x\rightarrow 0-}(\frac{1}{x}-\frac{1}{|x|}) =-\infty)$

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