Limit of 1/x, as x approaches inf, question.

  • Thread starter ellis818
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  • #1
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Main Question or Discussion Point

Multiplication Question

If we know that [tex]\frac{a}{x}[/tex] = c, we than say a=x*c

We also know that [tex]\lim_{x\rightarrow \infty}[/tex] [tex]\frac{a}{x}[/tex] = 0

But what if we said: [tex]\frac{a}{x}[/tex] = c, then [tex]\frac{a*x}{x}[/tex] = x*c

So, if we take a limit: [tex]\lim_{x\rightarrow \infty}[/tex] [tex]\frac{a*x}{x}[/tex] = a

And also take a limit of: [tex]\lim_{x\rightarrow \infty}[/tex] x*c = [tex]\infty[/tex]

We get a problem.
 
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Answers and Replies

  • #2
rock.freak667
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If we know that [tex]\frac{a}{b}[/tex] = c
Could you make your question a bit more specific?
 
  • #3
jgens
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Is the question supposed to be: lim (x -> infinity) 1/x?
 
  • #4
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Sorry, accidently hit the Post button without being finished.
 
  • #5
jgens
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I think that the resolution of the conflict arises in the fact that lim (x -> infinity) cx = infinity when c is not equal to zero. I think in this particular case lim (x -> infinity) cx is an indeterminate form in the form of 0*infinity. I could be entirely wrong though.
 
  • #6
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The problem seems to be that c depends on x. Remember, c = x / a, so
[tex]\lim_{x \to \infty} cx = \lim_{x \to \infty} \frac{x}{a} x = \lim_{x \to \infty} a = a.[/tex]
 
  • #7
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[tex]\lim_{x \to \infty} cx = \lim_{x \to \infty} \frac{x}{a} x = \lim_{x \to \infty} a = a.[/tex]
Could you please explain this a bit further?
 
  • #8
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Whoops, I made a typo. It should be c = a / x and

[tex]\lim_{x \to \infty} cx = \lim_{x \to \infty} \frac{a}{x} x = \lim_{x \to \infty} a = a.[/tex]
 
  • #9
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Actually what I think you meant was, [tex]\lim_{x \to \infty} \frac{a}{x} x[/tex]
 
  • #10
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adriank I understand what you did, but the question I have is that both sides are not valid. I wasn't looking for a substatution of c being a/x.
 
  • #11
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I'm using what you used for c in your original post. The point is, in your post, c is not a constant.
 
  • #12
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That's what I was missing. Thanks adriank.
 
  • #13
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This leads me to another question, somewhat relevant to the original post.

If [tex]\lim_{x \to \infty} x (\lim_{x \to \infty}\frac{a}{x})[/tex] = a

Does this sort of mean that [tex]\infty[/tex]*0 = a ?

a is any real number except 0
 
  • #14
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This leads me to another question, somewhat relevant to the original post.

If [tex]\lim_{x \to \infty} x (\lim_{x \to \infty}\frac{a}{x})[/tex] = a

Does this sort of mean that [tex]\infty[/tex]*0 = a ?

a is any real number except 0
i would say that [tex]\lim_{x \to \infty} x (\lim_{x \to \infty}\frac{a}{x})[/tex] is an intermediate form, that is [tex]\infty*0[/tex] so as it is right now, we cannot evaluate the limit, but if we transform it into

[tex]\lim_{x \to \infty} x (\lim_{x \to \infty}\frac{a}{x})=\lim_{x\rightarrow \infty}x\frac{a}{x}=a[/tex]

Remember that infinity is not a number, so the laws that hold for real numbers, not necessarly will hold when infinity is in play.
 
  • #15
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Well basically my argument was the if we take the seperate parts of the two being:

[tex]\lim_{x \to \infty} x = \infty [/tex]

and

[tex]\lim_{x \to \infty}\frac{a}{x} = 0 [/tex]

multiply them together and get

[tex]\lim_{x \to \infty}\frac{a}{x}x = a [/tex]
 
  • #16
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This leads me to another question, somewhat relevant to the original post.

If [tex]\lim_{x \to \infty} x (\lim_{x \to \infty}\frac{a}{x})[/tex] = a

Does this sort of mean that [tex]\infty[/tex]*0 = a ?

a is any real number except 0
Your equation is ambiguous; there are two possible interpretations. First, and I'm guessing this is what you meant:
[tex]\left(\lim_{x \to \infty} x\right) \left(\lim_{x \to \infty}\frac{a}{x}\right) = a.[/tex]
The first limit doesn't exist, so that doesn't work. (Infinite limits are said to not exist; infinity is not a number. You can't multiply by it.)

Remember that in general,
[tex]\lim_{x \to \infty} f(x) g(x) = \left( \lim_{x \to \infty} f(x) \right) \left( \lim_{x \to \infty} g(x) \right)[/tex]
is true only if each of the limits exists.


Second:
[tex]\lim_{x \to \infty} \left(x \lim_{x \to \infty}\frac{a}{x}\right) = a.[/tex]
That just doesn't make sense. It isn't meaningful. You can't use the same variable in a limit that's inside another limit; it's just wrong to write that.

Perhaps you meant this:
[tex]\lim_{y \to \infty} \left(y \lim_{x \to \infty}\frac{a}{x}\right).[/tex]
In that case, evaluate the limits one at a time:
[tex]\lim_{y \to \infty} \left(y \lim_{x \to \infty}\frac{a}{x}\right)
= \lim_{y \to \infty} \left(y \cdot 0\right)
= \lim_{y \to \infty} 0 = 0.[/tex]
 
  • #17
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Infinite limits are said to not exist
Your right. I should have noticed that. I guess I was lazy to check that idea in the first place.

Thanks again, adriank.
 

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