Question

Let \( f : [-\pi, \pi] \to \mathbb{R} \) be a continuous function such that \( f(x)>\frac{f(0)}{2} \) for \( |x|<\delta \), where \( 0<\delta<\pi \). Define \( P_{n,\delta}(x) = (1 + \cos x - \cos \delta)^n \), for \( n = 1, 2, 3, \dots \). Then, which of the following statements is TRUE?

• A. \( \lim_{n \to \infty} \int_0^{2\delta} f(x) P_{n,\delta}(x) \, dx = 0 \)
• B. \( \lim_{n \to \infty} \int_{-\delta}^{0} f(x) P_{n,\delta}(x) \, dx = 0 \)
• C. \( \lim_{n \to \infty} \int_{-\delta}^{\delta} f(x) P_{n,\delta}(x) \, dx = 0 \)
• D. \( \lim_{n \to \infty} \int_{[-\pi, \pi] \setminus [-\delta, \delta]} f(x) P_{n,\delta}(x) \, dx = 0 \)

Hint:

When dealing with integrals involving functions that approach zero as \( n \) increases, focus on the behavior of the integrand as \( n \) becomes large. In this case, the rapid decay outside a small neighborhood around zero ensures that the integral over larger regions vanishes.

Answer 1

The Correct Option is D

Step 1: Analysis of \( P_{n,\delta}(x) \)
The function \( P_{n,\delta}(x) = (1 + \cos x - \cos \delta)^n \) tends to 0 for most values of \( x \) except near \( x = 0 \). As \( n \to \infty \), the integrand becomes negligible except for values of \( x \) near zero. Step 2: Contribution of regions far from 0
For large \( n \), the contribution from regions where \( |x| \) is not close to zero rapidly vanishes, because \( P_{n,\delta}(x) \) decays quickly outside a small neighborhood around \( x = 0 \). Step 3: Conclusion
Thus, the integral over the region \( [-\pi, \pi] \setminus [-\delta, \delta] \) vanishes as \( n \to \infty \). Therefore, the correct answer is (D).