For real numbers \( a, b \), find the following limit.

\[

\lim_{n \to \infty} n \left( 1 – \frac{a}{n} – \frac{b \log (n+1)}{n} \right)^n.

\]

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For real numbers \( a, b \), find the following limit.

\[

\lim_{n \to \infty} n \left( 1 – \frac{a}{n} – \frac{b \log (n+1)}{n} \right)^n.

\]

Let \( x, y \) be real numbers satisfying \( y \geq x^2 + 1 \). Prove that there exists a bounded random variable \( Z \) such that

\[

E[Z] = 0, E[Z^2] = 1, E[Z^3] = x, E[Z^4] = y.

\]

Here, \( E \) denotes the expectation.

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