×
Back to all chapters

# 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.

# Limits of Functions: Level 5 Challenges

\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...

×