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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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