Limits of Functions

Limits of Functions: Level 5 Challenges


limn(n0)(n1)(n2)(nn)n2+n \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 (sn)n=0(s_n)_{n=0}^{\infty} be a sequence of real numbers defined as follows:

s0=2;sn+1=24sn2s_0 = 2; s_{n+1} = \sqrt{2-\sqrt{4-s_n^2}} for n0n \ge 0.

To the nearest hundredth, find the value of limn2nsn\displaystyle\lim_{n \to \infty} 2^n s_n.

In other words, to what value does the following sequence converge:23s3=822+22^3 s_3 = 8\sqrt{2-\sqrt{2+\sqrt{2}}}24s4=1622+2+22^4 s_4 = 16\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2}}}}25s5=3222+2+2+22^5 s_5 = 32\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}and so on...

limn(n1)(n2)(nn)n2\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: (nk)=n!k!(nk)!\displaystyle {n \choose k} = \dfrac {n!}{k!(n-k)!} denotes the binomial coefficient.

For a=116a=\dfrac1{16}, consider the (finite) power tower,

xn=aaaa2n  a’s\Large x_n=\underbrace{a^{a^{\cdot^{\cdot^{a^a}}}}}_{2n \; a\text{'s}}

For example, x1=aax_1=a^a and x2=aaaax_2=a^{a^{a^a}}.

Find limnxn\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 aa's?

f(x)={0 if x is irrational1 if x is rationalg(x)={0 if x is irrational1q if x=pq, where p and q are coprime nonnegative integers \begin{aligned} 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{aligned}

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

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

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


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