To evaluate the limit by recognizing it as a Riemann sum for a function defined on the interval [0,1], we first need to understand what a Riemann sum represents. A Riemann sum approximates the integral of a function over a specified interval by dividing that interval into smaller subintervals.
The limit in question often appears in the form:
\[ ext{Limit}
ightarrow rac{1}{n} imes ext{sum of a function evaluated at each partition} \]
1. **Identify the function:** Consider the function you want to evaluate. Let’s take a general function, say \( f(x) \), defined on the interval [0, 1]. We need to express our limit as a sum that can fit the Riemann sum definition.
2. **Divide the interval:** We divide the interval [0, 1] into \( n \) equal parts. Each subinterval has a width of \( \\Delta x = \frac{1}{n} \). The points of division can be defined as \( x_i = \frac{i}{n} \) for \( i = 0, 1, 2, \ldots, n \).
3. **Write the Riemann sum:** The Riemann sum can be expressed as:
\[ S_n = \sum_{i=1}^{n} f(x_i^*) \\Delta x \]
where \( x_i^* \) is a sample point in the i-th subinterval. In many cases, we take \( x_i^* = x_i \).
4. **Set up the limit:** Suppose the limit you are trying to evaluate resembles the following form:
\[ L = \lim_{n \to \infty} S_n = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i) \frac{1}{n} \]
If your function \( f(x) \) is continuous on the interval [0, 1], then as \( n \) approaches infinity, the Riemann sum converges to the definite integral:
\[ L = \int_{0}^{1} f(x) \, dx \]
5. **Compute the integral:** Now, compute the integral using the appropriate method (substitution, integration by parts, numerical methods, etc.) to find the limit.
In summary, by identifying the limit as a Riemann sum, dividing the interval, setting up the sum, and then computing the corresponding integral, we can effectively evaluate the limit in a structured way.