Calculus

# Limits of Functions: Level 4 Challenges

Let $$\{x_n\}$$ be a sequence such that $$x_1=1,\ x_nx_{n+1}=2n$$ for $$n\ge 1$$.

Find $$\displaystyle \lim_{n\to\infty} \dfrac{|x_{n+1}-x_n|}{\sqrt{n}}.$$

$\lim_{n \rightarrow \infty} \underbrace{\cos \cos \cos \ldots \cos}_{n} \ x$

Evaluate the limit above. Give your answer rounded to three decimal places.

If you think the limit does not exist, submit your answer as 123.

Compute $\large \lim_{ n\to{\infty}}{\dfrac{ 2016(1^{2015}+2^{2015}+ \cdots +n^{2015}) - n^{2016}}{2016(1^{2014}+2^{2014}+\cdots+n^{2014})}}.$

$\large \displaystyle \lim_{n \to \infty} \dfrac {A_n}{G_n}$

Let $$a_1, a_2, \ldots, a_n$$ be an arithmetic progression composed solely of distinct positive reals.

Also, let $$A_n$$ denote its arithmetic mean and $$G_n$$ denote its geometric mean.

What is the value of the above limit to 3 decimal places?

If you think the limit doesn't exist, write -13.37 as your answer.

$\large \displaystyle \lim_{x\to\infty}\left(\sqrt[3]{8^x+3^x}-\sqrt{4^x-2^x} \right)= \, ?$

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