how to solve?---Question:
Extract the mathematical expression from the image.
Mathematical Expression:
lim y->inf y * integral_{0}^{1} x^y f(x) dx = f(1)
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We examine a fundamental limit theorem involving weighted integrals. The theorem states that the limit as y approaches infinity of y times the integral from zero to one of x to the y power times f of x dx equals f of one. This result shows how the weight function x to the y power concentrates mass near x equals one as y increases.
Understanding the weight function x to the y power is crucial. As y increases, this function exhibits concentration behavior. For x less than one, x to the y approaches zero. For x equal to one, x to the y always equals one. For x greater than one, x to the y approaches infinity. This creates a spike at x equals one, concentrating the integral weight near the right endpoint. The factor y compensates for this narrowing, leading to the limit f of one.
The proof strategy involves three key steps. First, we decompose the integral into regions near x equals one and away from x equals one. For the region near x equals one, we use the continuity of f to approximate f of x by f of one. For the region away from x equals one, we show that y times the integral of x to the y times f of x approaches zero. The combination of these steps yields the desired limit f of one.
This limit theorem has important applications in various fields. In probability theory, it relates to weak convergence of probability measures and moment problems. In approximation theory, it connects to Bernstein polynomials and Weierstrass approximation. In numerical analysis, it helps with quadrature rules and integral estimation. For example, when f of x equals x squared, the limit gives us that y times the integral from zero to one of x to the power y plus two dx approaches one as y approaches infinity.
To summarize what we have learned: The limit theorem shows how weighted integrals concentrate at boundary points. The weight function x to the y creates a spike at x equals one as y approaches infinity. The factor y compensates for the narrowing weight distribution. This result has applications in probability theory, approximation theory, and numerical analysis. The theorem demonstrates the power of limit analysis in understanding integral behavior.