# 2016-2 Integral limit

For $$a \geq 0$$, find
$\lim_{n \to \infty} n \int_{-1}^0 \left( x + \frac{x^2}{2} + e^{ax} \right)^n dx.$

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# 2015-11 Limit

Does $$\frac{1}{n \sin n}$$ converge as $$n$$ goes to infinity?

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# 2014-22 Limit

For a nonnegative real number $$x$$, let $f_n(x)=\frac{\prod_{k=1}^{n-1} ((x+k)(x+k+1))}{ (n!)^2}$ for a positive integer $$n$$. Determine  $$\lim_{n\to\infty}f_n(x)$$.

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# 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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# 2012-19 A limit of a sequence involving a square root

Let $$a_0=3$$ and $$a_{n}=a_{n-1}+\sqrt{a_{n-1}^2+3}$$ for all $$n\ge 1$$. Determine $\lim_{n\to\infty}\frac{a_n}{2^n}.$

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# 2012-5 Iterative geometric mean

For given positive real numbers $$a_1,\ldots,a_k$$ and for each integer n≥k, let $$a_{n+1}$$ be the geometric mean of $$a_n, a_{n-1}, a_{n-2}, \ldots, a_{n-k+1}$$. Prove that $$\lim_{n\to\infty} a_n$$ exists and compute this limit.

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# 2011-23 Constant Function

Let $$f:\mathbb{R}^n\to \mathbb{R}^{n-1}$$ be a function such that for each point a in $$\mathbb{R}^n$$, the limit $$\lim_{x\to a} \frac{|f(x)-f(a)|}{|x-a|}$$ exists. Prove that f is a constant function.

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# 2011-8 Geometric Mean

Let f be a continuous function on [0,1]. Prove that $\lim_{n\to \infty}\int_0^1 \cdots \int_0^1 f(\sqrt[n]{x_1 x_2 \cdots x_n } ) dx_1 dx_2 \cdots dx_n = f(1/e).$

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# 2010-21 Limit

Let $$a_1=0$$, $$a_{2n+1}=a_{2n}=n-a_n$$. Prove that there exists k such that $$\lvert a_k- \frac{k}{3}\rvert >2010$$ and yet $$\lim_{n\to \infty} \frac{a_n}{n}=\frac13$$.

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Let f be a differentiable function. Prove that if $$\lim_{x\to\infty} (f(x)+f'(x))=1$$, then $$\lim_{x\to\infty} f(x)=1$$.