To solve this problem, we should first notice that the given limit is in the form of a Riemann sum, which corresponds to the integral of a function from a specific interval A to B. The Riemann sum of a function f(x) over the interval [A, B] is obtained by dividing that interval into many small subintervals (the number of which tends to infinity), approximating the function by a constant over each subinterval, and summing the areas of the rectangles thus obtained.
So, to find the function f(x), we replace n with x and i/n with t in the expression inside the summation of the original limit given:
n/(4 + n * i/n) => x/(4 + x*t)
Then, after simplifying both sides, we obtain:
1/(4/x + t)
This is our function f(x), i.e., f(x) = 1/(4/x + t).
Now, we need to find the interval (A, B) for the integral. In terms of the limit, this corresponds to the sum going from i = 1 to n as n tends to infinity.
In terms of the integral, these correspond to the values of the function's independent variable t, over which the integral is evaluated. Since i was divided by n to give us t, as n (which is essentially representing the number of subdivisions of the interval on which the function is defined) goes to infinity, 1/n goes to 0 and n/n goes to 1.
Hence the interval is from t = 0 to t = 1, or in other terms, A = 0 and B = 1.
To summarize, the function f(x) is f(x) = 1/(4/x + t) and the interval for the integral representation of the limit is A = 0, B = 1.