# What is 0 multiplied by infinity in limits?

## Main Question or Discussion Point

If you have f(x) = 1/x and g(a) = (cosx - 1)/x and then y = [limx→0 f(x)][limx→0g(x)], the two individual limits equal 0 and infinity, respectively. Since these are limits and only approach these values, would the multiplication of the two limits equal 0, infinity or something else?

Office_Shredder
Staff Emeritus
Gold Member
In this case, $$y= \lim_{x\to 0} f(x) \lim_{x\to 0} g(x)$$ is a meaningless expression. You CAN calculate
$$\lim_{x\to 0} f(x) g(x)$$
but you cannot split it into two limits and give the new expression any meaning.

Mark44
Mentor
If you take a limit and get 0 * ∞, it means that you're not done yet. The indeterminate form [0 * ∞] is one of several indeterminate forms that can arise when you're taking limits. One thing that is often done is to rewrite the expression so that the limit becomes [0/0] or [∞/∞], either of which might be amenable to evaluation using L'Hopital's Rule.

This wiki article has a table of indeterminate forms - http://en.wikipedia.org/wiki/Indeterminate_form#List_of_indeterminate_forms

SteamKing
Staff Emeritus
Homework Helper
If you have f(x) = 1/x and g(a) = (cosx - 1)/x and then y = [limx→0 f(x)][limx→0g(x)], the two individual limits equal 0 and infinity, respectively. Since these are limits and only approach these values, would the multiplication of the two limits equal 0, infinity or something else?
The limit of f(x) = 1/x as x approaches 0 is not zero, it is infinity.

Mark44
Mentor
The limit of f(x) = 1/x as x approaches 0 is not zero, it is infinity.
You're sort of half correct.
$$\lim_{x \to 0^+}\frac 1 x = ∞$$
$$\lim_{x \to 0^-}\frac 1 x = -∞$$
Since the one-sided limits aren't equal, the two-sided limit doesn't exist.

Sorry for my poor phrasing.

How about in this new example.

y = [limx→0^+ 1/x][limz→0 (1 - cosz)/z]

Would it be possible to evaluate this limit since one limit approaches infinity while the other approaches 0, and they are different variables in this case.

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Mark44
Mentor
Sorry for my poor phrasing.

How about in this new example.

y = [limx→0^+ 1/x][limz→0 (1 - cosz)/z]

Would it be possible to evaluate this limit since one limit approaches infinity while the other approaches 0, and they are different variables in this case.
No, you're right back to what I was talking about in post #3, and what you had in your first post. The fact that you have different variables does not change things.

Mark44
Mentor
How did this limit come up? What's the problem you're trying to solve?