# 2013-16 Limit of a sequence

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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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.