Answer:
2
Step-by-step explanation:
A=r²
(19/6)=(285/360)r²
Divide both sides by pi, and they cancel out.
x is the radius, so replace r with x.
19/6=285/360x²
19/6÷285/360=x²
(19/6)·(360/285)=x²
4=x²
√4=x
2=x
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.