Evaluate $\lim_{x \to 0^-} \left( \frac{1}{x} - \frac{1}{|x|} \right)$ (possible textbook mistake - James Stewart 7th) I was working on a few problems from James Stewart's Calculus book (seventh edition) and I found the following:

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

Since there's a $|x|$ on the limit and knowing that $|x| = -x$ for any value less than zero, we have
$$\lim_{x \to 0^-} \left( \frac{1}{x} - \frac{1}{|x|} \right) = \lim_{x \to 0^-} \frac{2}{x}$$
So far so good. Continuing,
$$\lim_{x \to 0^-} \left( \frac{1}{x} - \frac{1}{|x|} \right) = \lim_{x \to 0^-} \frac{2}{x} = - \infty$$
since the denominator becomes smaller and smaller. When checking the textbook's answer I've found the following:


Am I missing something or should the limit really be $- \infty$ ?
 A: Saying that the limit is equal to $\infty$ is a mathematical shorthand (amongst some mathematicians, at least) for: 
Given any real number $M$, there is a real $\delta$ (depending on $M$) such that $\frac{1}{x^2} > M $ for all $x$ satisfying $0 <x <|\delta|$. 
It is usually advised that beginners avoid using $\infty$ since it leads to careless or wrong manipulations of the symbol all too often. 
A: bru, I think that the problem is just about terminology. Your derivation is correct, but it is likely that what Stewart is claiming is (I guess) that a limit that goes to $-\infty$ or to $+\infty$ on only one side (as in this example, where the limit is only from the left), is "non existing". Otherwise, the statement "the limit does not exist becuase the denomiator approaches $0$ while the numerator does not" would be simply nonsense.  
A: I agree that the limit is negative infinity. I could understand "does not exist" if the question was asking for a two-sided limit, but it clearly isn't. It's asking for the left-hand limit. The text differentiates between specifically identifying ± infinite limits and non-existent limits, so it should be consistent in the solution manual. I just burned an hour trying to figure out if this was some strange concept that I missed... and I'm the teacher! Glad I found this thread!
