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

## Main Question or Discussion Point

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.

Last edited:

## Answers and Replies

rock.freak667
Homework Helper
If we know that $$\frac{a}{b}$$ = c
Could you make your question a bit more specific?

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

Sorry, accidently hit the Post button without being finished.

jgens
Gold Member
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.

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

$$\lim_{x \to \infty} cx = \lim_{x \to \infty} \frac{x}{a} x = \lim_{x \to \infty} a = a.$$
Could you please explain this a bit further?

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

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

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

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.

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

That's what I was missing. Thanks 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

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.

Well basically my argument was the if we take the seperate 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

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

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:
$$\lim_{y \to \infty} \left(y \lim_{x \to \infty}\frac{a}{x}\right) = \lim_{y \to \infty} \left(y \cdot 0\right) = \lim_{y \to \infty} 0 = 0.$$

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.