0

Find points of discontinuity of

$$f(x)=\lim\frac{\left(1+\sin (π/x)\right)^n-1}{\left(1+\sin (π/x)\right)^n+1}, \,x \in (0,1).$$

My Attempt:

When $x$ is irrational, $0<x<1 \implies 1<1/x \implies \pi <\pi/x \implies 0< \sin(\pi/x) \implies 1<1+\sin(\pi/x),$

so that

$$f(x)=\lim\frac{1- 1/(1+\sin \pi/x)^n}{1+ 1/(1+\sin \pi/x)^n},$$ that is $f(x)=1$ as $$\lim 1/(1+\sin \pi/x)^n=0.$$

Again when $x$ is rational $f(x)=0,$ so $f$ is discontinuous on $(0,1).$

Am I right?

Allawonder
  • 13,327
choton choton
  • 11
  • 2
    I don't completely understand what you wrote. For the example, with your "f(x)=lim{(1+sin pi/x)^n-1}/{(1+sin pi/x)^n+1}". You have a "lim", but don't state what the limit you're taking is. It would appear it should be for $n \to \infty$. Also, I don't understand what $x$ being irrational or rational has to do with it, including what you're saying about their results. One thing which would help is to use MathJax to format your math statements, with this tuturial helping to explain the basics. – John Omielan Sep 19 '19 at 03:34
  • next time surely i will take help from MathJax – choton choton Sep 19 '19 at 05:56
  • yes i wnt to ask the question which is edited – choton choton Sep 19 '19 at 06:06
  • when x is rational argument of sin is n.pi so f(X)=0 – choton choton Sep 19 '19 at 06:12
  • if i am in wrong direction plz correct me by solving the problem – choton choton Sep 19 '19 at 06:13

0 Answers0