What is lim(x->0)1/x=0?

  • Thread starter coldboyqn
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  • #1
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I am doing something that I sure that I'm wrong, but I cannot realize the error. See as below:
[tex]
\frac{1}{x}=x\times\frac{1}{(x^2)} ________\(1\)
[/tex]

Taylor Series of [tex]\frac{1}{x^2}[/tex]:
[tex]
\frac{1}{x^2}=\frac{1}{\alpha}+\sum_{k=1}^\infty g(k)(x-\alpha)^k
[/tex]

In which
k is from 1 to infinity,
[tex]
g(k)=(-1)^k\times\frac{(k+1)!}{\alpha^{(k+2)}k!}\\\\
=\frac{(-1)^k\times(k+1)}{\alpha^{(k+2)}}
[/tex]

Substitute Taylor Series of 1/x^2 into (1), we obtain:
[tex]\frac{1}{x}=\frac{x}{\alpha}+\sum_{k=1}^\infty g(k)(x-\alpha)^k[/tex]
So: [tex]\lim_{\substack{x\rightarrow 0}} \frac{1}{x}=\lim_{\substack{x\rightarrow 0}} (\frac{x}{\alpha}+\sum_{k=1}^\infty x\times g(k)(x-\alpha)^k)=0 (!?!?!?)
[/tex]
Can any one show me, please?
 
Last edited:

Answers and Replies

  • #2
StatusX
Homework Helper
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You need to check the radius of convergence of the taylor series. Note if you don't multiply by x, you get the limit as x->0 of 1/x^2 to be 1/a^2, where a is arbitrary. (I'm assuming alpha=a^2)
 
  • #3
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I see that if using the Taylor series above to determine the value of [tex]\lim_{\substack{x\rightarrow 0}}\frac{1}{x^2}[/tex] we will obtain infinity, which is according?
 
  • #4
StatusX
Homework Helper
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You're right. Sorry, I should have read your question more carefully. The problem is that the last limit isn't 0. You can't just plug in 0 to get the limit, as the function isn't continuous at x=0. More careful calculation should show that limit diverges as well.
 
  • #5
JasonRox
Homework Helper
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Wouldn't it be best to draw a graph? Especially when it's possible.
 
  • #6
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StatusX said:
The problem is that the last limit isn't 0. You can't just plug in 0 to get the limit, as the function isn't continuous at x=0.
Surely the last limit isn't 0. But my problem is that the strange result I obtain when
I treat [tex]\frac{1}{x}[/tex] as [tex]x\times TaylorSeries\_of(\frac{1}{x^2})[/tex].
I wonder where is my error when I calculate the limit by this method!
 
  • #7
StatusX
Homework Helper
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It isn't hard to check the series converges precisely in (0,2a), and in this region it converges to 1/x^2. If you multiply it termwise by x, you get a series that converges in [0,2a). But there's no reason to expect that series evaluated at x=0 to give you the same thing as (1/x^2)*x evaluated at 0, since the taylor series did not converge at x=0. You can prove the limit diverges explicitly, and the easiest way to do this is just to prove the series does converge to 1/x in (0,2a).
 
  • #8
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Oh, yes, I see. As the Taylor Series of [tex]\frac{1}{x^2}[/tex] diverges at 0, I cannot simply multiply it with x to evaluate 1/x at 0, right? And it is unreasonable to multiply an expression that diverges (to [tex]\infty[/tex]) with a variable that come to zero and conclude that the multiplied expression come to zero, right?
Thanks for explanation, I understand now.
 

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