To evaluate the limit, we start by rewriting the sum in question as a Riemann sum. A Riemann sum approximates the integral of a function over an interval using finite sums. This can be particularly useful in calculus when determining integrals.
Consider the limit:
lim (n -> ∞) Σ f(x_i) Δx
where:
- Δx = (b – a) / n, which is the width of each subinterval. In our case, the interval is [0, 1], so Δx = 1/n.
- x_i = a + iΔx, which gives us the sample points in the interval. For our interval, x_i = 0 + (i/n) = i/n.
Now, if we recognize our sum as:
Σ f(x_i) Δx = Σ f(i/n) (1/n)
As n approaches infinity, this sum approaches the integral of the function f over the interval [0, 1]. Thus, we can express the limit of the sum as:
lim (n -> ∞) Σ f(i/n) (1/n) = ∫01 f(x) dx
In conclusion, to evaluate the limit, we first need to rewrite the sum in the form of a Riemann sum, then recognize that as n approaches infinity, this will converge to the definite integral of the function over the specified interval. This gives us a powerful tool to tackle problems involving limits and sums in the context of calculus.