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Limits of Functions

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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