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3. (4 marks) Justify the following equalities: (a) $\lim_{n \to \infty} \int_0^1 [4x(1 - x)]^n e^{\sin(nx)} dx = 0$. (Hint: $|4x(1 - x)| < 1$ for all $x \in [0, 1] \setminus \{\frac{1}{2}\}$). (b) $\lim_{n \to \infty} \int_0^1 n \ln \left(1 + \left(\frac{x}{n}\right)^\alpha \right) dx = \begin{cases} 0 & \text{if } \alpha > 1\\ \frac{1}{2} & \text{if } \alpha = 1 \end{cases}$

          3. (4 marks) Justify the following equalities:
(a) $\lim_{n \to \infty} \int_0^1 [4x(1 - x)]^n e^{\sin(nx)} dx = 0$.
(Hint: $|4x(1 - x)| < 1$ for all $x \in [0, 1] \setminus \{\frac{1}{2}\}$).
(b) $\lim_{n \to \infty} \int_0^1 n \ln \left(1 + \left(\frac{x}{n}\right)^\alpha \right) dx = \begin{cases} 0 & \text{if } \alpha > 1\\ \frac{1}{2} & \text{if } \alpha = 1 \end{cases}$

$3. (4 marks) Justify the following equalities: (a) limn →∞∫0^1 [4x(1 - x)]^n e^sin(nx) dx = 0. (Hint: |4x(1 - x)| < 1 for all x ∈ [0, 1] ∖{(1)/(2)}). (b) limn →∞∫0^1 n ln(1 + ((x)/(n))^α) dx = 0 if α > 1 (1)/(2) if α = 1$

Added by Abigail B.

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