Limits of Functions

Limits of Functions: Level 4 Challenges


limn(nsinπn)(1+πn)n \Large \lim_{n\to\infty} \left( n \sin \dfrac \pi n\right)^{\left( 1 + \frac \pi n\right)^n}

The limit above has a closed form. Find the value of this closed form.

If this limit can be approximated as 3.19×10η3.19\times 10^\eta, where η\eta is an integer, find η\eta.

You may use a calculator for the final step of your calculation.

limxa(2ax)tanπx2a=en\large \lim_{x \to a} \bigg( 2 - \frac{a}{x} \bigg)^{\tan \frac{\pi x}{2a}} = e^{n}

Find approximate value of nn.

limn(a+bn1a)n \large \displaystyle \lim_{n \to \infty} \left(\dfrac{a+\sqrt[n]{b}-1}{a}\right)^{n}

Calculate the limit above in terms of aa and bb, where aa and bb are constants with b0b\geq0 .

Let f:(1,)(0,)f:(1,\infty) \rightarrow (0,\infty) be a continuous decreasing function with limxf(4x)f(8x)=1 \large \lim_{x\to\infty} \dfrac{f(4x)}{f(8x)} = \, 1


limxf(6x)f(8x)=? \large \lim_{x\to\infty} \dfrac{f(6x)}{f(8x)} = \, ?

Let {xn}\{x_n\} be a sequence such that x1=1, xnxn+1=2nx_1=1,\ x_nx_{n+1}=2n for n1n\ge 1.

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

Compute limn2016(12015+22015++n2015)n20162016(12014+22014++n2014). \large \lim_{ n\to{\infty}}{\dfrac{ 2016(1^{2015}+2^{2015}+ \cdots +n^{2015}) - n^{2016}}{2016(1^{2014}+2^{2014}+\cdots+n^{2014})}}.


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