Express the limit as a definite integral on the given interval.lim n→∞ ni = 1xi*(xi*)2 + 8Δx,    [1, 3]1  dx

The given limit is a Riemann sum for a definite integral over the interval [1, 3]. The function to be integrated is f(x) = x/(x^2 + 8).

The Riemann sum is given by:

lim n→∞ Σ (from i=1 to n) f(xi*) Δx

where xi* is a sample point in the i-th subinterval, and Δx = (b - a)/n is the width of each subinterval in a partition of [a, b].

In this case, a = 1, b = 3, and xi* = xi = i/n.

As n approaches infinity, the Riemann sum approaches the definite integral of f(x) from a to b.

Therefore, the limit can be expressed as the definite integral:

∫ (from 1 to 3) x/(x^2 + 8) dx