The answer is the 3rd question
Answer:
4
Step-by-step explanation:
Add them together and find the area of both
By the Stolz-Cesaro theorem, this limit exists if
also exists, and the limits would be equal. The theorem requires that
be strictly monotone and divergent, which is the case since
.
You have
so we're left with computing
This can be done with the help of Stirling's approximation, which says that for large
,
. By this reasoning our limit is
Let's examine this limit in parts. First,
As
, this term approaches 1.
Next,
The term on the right approaches
, cancelling the
. So we're left with
Expand the numerator and denominator, and just examine the first few leading terms and their coefficients.
Divide through the numerator and denominator by
:
So you can see that, by comparison, we have
so this is the value of the limit.