The limit of the product is the product of the limits.
.
as long as all limits involved exist, this "limit of a product is the product of the limits'' also works even when the product involves several factors.
Example 2
Assume that exists, , and .
What is ?
Since all limits involved exist,
.
Example 3
Assume that exists, , and .
What is ?
Since exists, so does . Then we can say
.
Example 4
Assume that exists and .
What is ?
Since all the limits involved exist, we can pull this apart. First we split the limit across the sum in the numerator, then we'll worry about products.