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(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

i need more details sorry

step-by-step explanation: