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

[ellis818]

Multiplication Question

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

We also know that \lim_{x\rightarrow \infty} \frac{a}{x} = 0

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

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

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

We get a problem.

 

[rock.freak667]

If we know that \frac{a}{b} = c

Could you make your question a bit more specific?

 

[jgens]

Is the question supposed to be: lim (x -> infinity) 1/x?

 

[ellis818]

Sorry, accidently hit the Post button without being finished.

 

[jgens]

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.

 

[adriank]

The problem seems to be that c depends on x. Remember, c = x / a, so
\lim_{x \to \infty} cx = \lim_{x \to \infty} \frac{x}{a} x = \lim_{x \to \infty} a = a.

 

[ellis818]

\lim_{x \to \infty} cx = \lim_{x \to \infty} \frac{x}{a} x = \lim_{x \to \infty} a = a.

Could you please explain this a bit further?

 

[adriank]

Whoops, I made a typo. It should be c = a / x and

\lim_{x \to \infty} cx = \lim_{x \to \infty} \frac{a}{x} x = \lim_{x \to \infty} a = a.

 

[ellis818]

Actually what I think you meant was, \lim_{x \to \infty} \frac{a}{x} x

 

[ellis818]

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.

 

[adriank]

I'm using what you used for c in your original post. The point is, in your post, c is not a constant.

 

[ellis818]

That's what I was missing. Thanks adriank.

 

[ellis818]

This leads me to another question, somewhat relevant to the original post.

If \lim_{x \to \infty} x (\lim_{x \to \infty}\frac{a}{x}) = a

Does this sort of mean that \infty*0 = a ?

a is any real number except 0

 

[sutupidmath]

This leads me to another question, somewhat relevant to the original post.

If \lim_{x \to \infty} x (\lim_{x \to \infty}\frac{a}{x}) = a

Does this sort of mean that \infty*0 = a ?

a is any real number except 0

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

\lim_{x \to \infty} x (\lim_{x \to \infty}\frac{a}{x})=\lim_{x\rightarrow \infty}x\frac{a}{x}=a

Remember that infinity is not a number, so the laws that hold for real numbers, not necessarly will hold when infinity is in play.

 

[ellis818]

Well basically my argument was the if we take the separate parts of the two being:

\lim_{x \to \infty} x = \infty

and

\lim_{x \to \infty}\frac{a}{x} = 0

multiply them together and get

\lim_{x \to \infty}\frac{a}{x}x = a

 

[adriank]

This leads me to another question, somewhat relevant to the original post.

If \lim_{x \to \infty} x (\lim_{x \to \infty}\frac{a}{x}) = a

Does this sort of mean that \infty*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:
\left(\lim_{x \to \infty} x\right) \left(\lim_{x \to \infty}\frac{a}{x}\right) = a.
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,
\lim_{x \to \infty} f(x) g(x) = \left( \lim_{x \to \infty} f(x) \right) \left( \lim_{x \to \infty} g(x) \right)
is true only if each of the limits exists.Second:
\lim_{x \to \infty} \left(x \lim_{x \to \infty}\frac{a}{x}\right) = a.
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:
\lim_{y \to \infty} \left(y \lim_{x \to \infty}\frac{a}{x}\right).
In that case, evaluate the limits one at a time:
\lim_{y \to \infty} \left(y \lim_{x \to \infty}\frac{a}{x}\right)<br /> = \lim_{y \to \infty} \left(y \cdot 0\right)<br /> = \lim_{y \to \infty} 0 = 0.

 

[ellis818]

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.