Multiplying and Dividing Limits
As long as both and exist,
In words, the limit of a product is the product of the limits, as long as the limits involved exist.
Here's a few more examples of this:
If we try to break this limit into pieces, we find
This is trouble, because
What do we find if we multiply ∞ by something (if we could do such a thing, which we can't because ∞ is not a number)? It would make reasonable sense to assume we'd get ∞ again.
What do we get if we multiply a number by 0? We get 0. What if we try to multiply ∞ by 0? That makes no sense at all.
It turns out that in this case we can't find the limit of the product via the product of the limits.
However, we can still find the original limit we were asked about.
If the limits of f and g exist and
Suppose and Then,
If , we can't use the rule we've been using to evaluate something like
If we try we find that
then we're out of luck.