Calculus
# Limits of Functions

\[ \begin{align} f(x) &= \begin{cases} 0 &\text{ if } x \text{ is irrational} \\ 1 &\text{ if } x \text{ is rational} \end{cases} \\ g(x) &= \begin{cases} 0 &\text{ if } x \text{ is irrational} \\ \frac1q &\text{ if } x =\frac{p}{q}, \text{ where } p \text{ and } q \text{ are coprime nonnegative integers} \end{cases} \end{align} \]

Let \( f(x) \) and \( g(x) \) be two functions defined on \( [0,1] \) by the formulas as described above.

For which \( a \in (0,1) \) does \( \lim\limits_{x\to a} f(x) \) exist?

For which \( b \in (0,1) \) does \( \lim\limits_{x\to b} g(x) \) exist?

\[\large \lim_{n\to\infty} \sqrt[n^2]{{n \choose1}{n \choose 2}\cdots{n \choose n}}\]

Find the closed form of the limit above to 3 decimal places.

**Notation:** \(\displaystyle {n \choose k} = \dfrac {n!}{k!(n-k)!}\) denotes the binomial coefficient.

For \(a=\dfrac1{16}\), consider the (finite) power tower,

\[\Large x_n=\underbrace{a^{a^{\cdot^{\cdot^{a^a}}}}}_{2n \; a\text{'s}} \]

For example, \(x_1=a^a\) and \(x_2=a^{a^{a^a}}\).

Find \(\displaystyle \lim_{n\to\infty}x_n\), to three significant figures.

**Bonus** What happens if we consider a power tower with an **odd** number of \(a\)'s?

\[ \large \lim_{n\rightarrow \infty} \, \sqrt[ \large n^{2}+n]{\binom{n}{0}\binom{n}{1}\binom{n}{2}\cdots \binom{n}{n}} \]

Find the value of the closed form of the above limit to 3 decimal places.

Let \((s_n)_{n=0}^{\infty}\) be a sequence of real numbers defined as follows:

\(s_0 = 2; s_{n+1} = \sqrt{2-\sqrt{4-s_n^2}}\) for \(n \ge 0\).

To the nearest hundredth, find the value of \(\displaystyle\lim_{n \to \infty} 2^n s_n\).

In other words, to what value does the following sequence converge:\[2^3 s_3 = 8\sqrt{2-\sqrt{2+\sqrt{2}}}\]\[2^4 s_4 = 16\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2}}}}\]\[2^5 s_5 = 32\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}\]and so on...

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