Finding the limit of the equation

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In summary, the limit \lim_{x\to 0} \frac{1}{x}-\frac{1}{|x|} does not exist as the left and right limits are not equal. This can be shown by using the sequential criterion for limits or by evaluating the limits from the positive and negative directions. Additionally, it is important to note that \frac{1}{2x} is not equal to \frac{1}{2}x.
  • #1
shwanky
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Homework Statement


Evaluate the limit if it exists

Homework Equations



[tex]\lim_{x\to 0} \frac{1}{x}-\frac{1}{|x|}[/tex]

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?
 
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  • #2
shwanky said:
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.
 
  • #3
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.
 
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  • #4
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?
 
  • #5
Yes. The limits from the two sides are different. So there is no common limit. So the limit doesn't exist.
 
  • #6
OMG! I can't believe it was that simple... :(
 
  • #7
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.
 
  • #8
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.
 
  • #9
[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!
 

1. What is the limit of an equation?

The limit of an equation is the value that a function approaches as the input approaches a specific value. It is denoted by the symbol "lim" and is used to describe the behavior of a function near a certain point.

2. How do you find the limit of an equation?

To find the limit of an equation, you can use several methods such as direct substitution, factoring, and L'Hopital's rule. The method used depends on the type of equation and the value that the input approaches.

3. Why is finding the limit of an equation important?

Finding the limit of an equation is important because it helps us understand the behavior of a function near a certain point. It allows us to determine if the function is continuous, has a discontinuity, or has a vertical asymptote. It also helps in solving problems involving rates of change and optimization.

4. What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit only considers the values of the function as the input approaches the specific value from one side (either from the left or right). A two-sided limit, on the other hand, considers the values of the function as the input approaches the specific value from both sides.

5. Can the limit of an equation exist even if the function is not defined at that point?

Yes, the limit of an equation can exist even if the function is not defined at that point. This is because the limit only considers the behavior of the function near the specific point and not the actual value of the function at that point.

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