Answer:
To find the value of n that makes the fraction A have the largest value, we can take the derivative of A with respect to n, set it equal to zero, and solve for n.
First, let's rewrite the fraction A as:
A = 1 + \frac{1}{n^{2}+1}
Next, let's take the derivative of A with respect to n:
dA/dn = 0 - 2n/(n^2+1)^2
Setting dA/dn equal to zero, we get:
0 = -2n/(n^2+1)^2
Multiplying both sides by (n^2+1)^2, we get:
0 = -2n
Therefore, n = 0 is a critical point of A. However, n cannot equal 0 because the denominator of A would be zero, which is undefined.
Instead, let's consider the limit of A as n approaches infinity. We have:
lim A = lim (1 + 1/(n^2+1)) = 1
Therefore, as n approaches infinity, A approaches 1.
On the other hand, as n approaches negative infinity, A approaches -1 because the denominator of A becomes negative while the numerator remains positive.
Therefore, the largest value of A occurs at n = infinity, and A approaches 1 as n approaches infinity.
In conclusion, the value of n that makes the fraction A have the largest value is infinity.