Answer:
See below
(B) and (C) are correct.
Step-by-step explanation:
We have the following limit
I am not sure about methods concerning the quotient, but in this type of question I would try to convert this limit into integration.
Considering the numerator, we have
- I didn't forget about
Considering the denominator, we have
- I didn't forget about
Therefore,
Now we have
This is just the notation change so far.
What I want to do here is apply definite integrals using Riemann Integrals (We will write the limit as an definite integral). A nice way to do it is using logarithms. Therefore, we can apply the natural logarithm in both sides.
Now, recall two properties of logarithms:
Thus,
Considering
Using Stirling's formula
then
This shows our limit equals 1 as and
Employing a Riemann sum in the main limit, we have
Now dividing the terms inside the parenthesis by in
we have
Now dividing the terms inside the parenthesis by in
we have
Therefore
for
Using Riemann Integral,
From
We can see that the function is increasing for , but because of the denominator, it is negative for .
Therefore,
(A) is false because
(B) is true because
(C) is true the slope is negative at that point
(D) is false, just consider for and