A Compilation of hard limits

Posted on Fri 05 June 2020 in Mathematics

This list consists of the limits that I found most challenging.

  1. $$\lim_{n \to \infty} \left( \frac{n!}{n^n} \right) ^\frac 1n$$
  2. $$\lim_{x \to 0} \left( \frac{1}{\ln(x + \sqrt{x^2+1})} - \frac 1{x+1} \right)$$
  3. $$\lim_{n \to \infty} \frac{n + n^2 + n^3 + ... + n^n}{1^n + 2^n + 3^n + ... + n^n}$$
  4. $$\lim_{n \to \infty} \left( \frac{n^n(x+n)\left(x+\frac n2\right)...\left(x+\frac nn\right)}{n!(x^2+n^2)\left(x^2+\frac {n^2}{4}\right)...\left( x^2 + \frac{n^2}{n^2}\right)}\right)^{\frac x n}$$
  5. $$\lim_{x \to 0} \left( 1^{\sin^{-2}x} + 2^{\sin^{-2}x} + ... + n^{\sin^{-2}x}\right)^{\sin^2 x}$$
  6. $$\lim_{n \to \infty} \sqrt[\leftroot{-2}\uproot{2}n+1]{(n+1)!}-\sqrt[\leftroot{-2}\uproot{2}n]{n!}$$
  7. $$\lim_{n \to \infty} \left( e - \left( 1 + \frac1n \right) ^n \right) ^\frac 1n$$
  8. $$\lim_{n \to \infty} \underbrace{\sin(\sin(\sin(...\sin(a)..)))}_\text{n times}$$
  9. $$\lim_{n \to \infty} \sqrt{1 + 2\sqrt{1 + 3\sqrt{1 + 4\sqrt{ ... 1 + (n-1)\sqrt{1+n}}}}}$$
  10. $$\lim_{x \to \infty} \binom{x}{n} \left(\frac{a}{x}\right)^n \left( 1 - \frac{a}{x}\right)^{x-n}$$
  11. $$\lim_{n \to \infty} \ \frac{2^{n+1}}{n+1} \ \left| 3\sum_{k=1}^n (-1)^k\frac{k}{2^k} + \frac 23\right|$$

Answers:

  1. $e$
  2. $-\frac 12$
  3. $1 - \frac 1e$
  4. $\exp\left( \int_0^x \ln \left( \frac{t+1}{t^2+1}\right)dt\right)$ (from JEE(A) 2016)
  5. $n$
  6. $e^{-1}$
  7. $1$
  8. $0$
  9. $3$ (Ramanujan found this one :)
  10. $\frac{a^n e^a}{n!}$
  11. $2$