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## Limits of Functions

What happens when a function's output isn't calculable directly – e.g., at infinity – but we still need to understand its behavior? That's where limits come in. See more

# Level 4

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

Find approximate value of $$n$$.

$\large \displaystyle \lim_{n\to\infty}10^{n+1}\left(x_n^{x_n} - y_n^{y_n}\right)$

Let $$x_n = 1.\underbrace{000000000\ldots 0}_{n \text{ number of 0's}}1$$ and $$y_n = 0.\underbrace{999999999\ldots9}_{(n+1) \text{ number of 9's}}$$.

Compute the limit above.

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

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

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

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

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

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

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

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

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