1. The problem statement, all variables and given/known data Evaluate the limit if it exists 2. Relevant equations [tex]\lim_{x\to 0} \frac{1}{x}-\frac{1}{|x|}[/tex] 3. The attempt at a solution 1) [tex]\lim_{x\to 0} \frac{1}{x}-\frac{1}{|x|}[/tex] 2a) [tex]\lim_{x\to 0} \frac{1}{x}-\frac{1}{x}[/tex] 2b) [tex]\lim_{x\to 0} \frac{1}{x}-\frac{1}{-x}[/tex] 3a) [tex]\lim_{x\to 0} 0 = 0[/tex] 3b) [tex]\lim_{x\to 0} \frac{1}{2x}[/tex] 4)[tex]\lim_{x\to 0} \frac{1}{2}x[/tex] 5)[tex] (\frac{1}{2})0 = 0[/tex] Did I do this correctly?
The limit is quite different if you approach zero from the negative direction or the positive direction. Split it into these two cases as I think you have tried to do, but 1a) 1b) 2a) 2b) etc is not the clearest way to express this. But finally 1/(2*x) is not equal to (1/2)*x. That's BAD.
Well, you didn't do it right as it doesn't exist. 1/x goes to -\infty as x->0 from the left. but -1/|x| is also going to -\infnty. Use the sequence criterion of limits for a more formal proof. Also note 1/x + 1/x is not 1/2x.
Thanks, I kind of figured it was wrong. I'm still a little shakey on limits. If it's a problem that I can factor and substitute, I'm okay, but my professor didn't clearly explain this in class. I can usually see that the limit doesn't exist, but I don't know how to state it. Should I just take the + and - limits and show that they don't approach a common point on an open interval as ZioX did?
Yes. The limits from the two sides are different. So there is no common limit. So the limit doesn't exist.
Define x_n=-1/n. This is a sequence converging to 0. But 1/x_n-1/|x_n|=-2n which doesn't converge to anything. Therefore 1/x-1/|x| has no limit as x tends to zero. This is the sequencial criterion for limits. A function f converges to L as x converges to a iff for every sequence x_n converging to a has the property that f(x_n) converges to L.
For the limit to exist, [tex]\lim_{x\to 0} \frac{1}{x}-\frac{1}{|x|} =\lim_{x\to 0^{+}} \frac{1}{x}-\frac{1}{|x|} = \lim_{x\to 0^{-}} \frac{1}{x}-\frac{1}{|x|}[/tex] However we can see that [tex] \lim_{x\to 0^{+}} \frac{1}{x}-\frac{1}{|x|}=0[/tex] but [tex]\lim_{x\to 0^{-}} \frac{1}{x}-\frac{1}{|x|} = \lim_{x\rightarrow 0} \frac{-2}{x}[/tex], which are obviously not the same.
[tex]\frac{1}{2x}\ne \frac{1}{2} x[/tex]!!! And, technically, you should say [itex]lim_{x\rightarrow 0^+}[/itex] and [itex]lim_{x\rightarrow 0^-}[/itex] but that isn't as bad as the howler above!