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ISI MStat 2018 PSA Problem 10 | Dirichlet Function

This is a problem from ISI MStat 2018 PSA Problem 10 based on Dirichlet Function.

Dirichlet Function - ISI MStat Year 2018 PSA Question 10


Let x be a real number. Then \lim {m \rightarrow \infty}\left(\lim {n \rightarrow \infty} \cos ^{2 n}(m ! \pi x)\right)

  • (A) does not exist for any x
  • (B) exists for all x
  • (C) exists if and only if x is irrational
  • (D) exists if and only if x is rational

Key Concepts


Limit

Sandwich Theorem

Check the Answer


Answer: is (B)

ISI MStat 2018 PSA Problem 10

Introduction to Real Analysis by Bertle Sherbert

Try with Hints


Check two cases separately one when x is rational and other is when x is irrational.

If m!x  is an integer, then cos ^{2 n}(m ! \pi x) =1

If x is rational \frac{p}{q}, then, eventually, for large enough m, m! will be divisible by q , so that m!x will be an integer, and we have \lim {m \rightarrow \infty}\left(\lim {n \rightarrow \infty} \cos ^{2 n}(m ! \pi x)\right) =1

If x is irrational, m!x  will never be an integer, and |cos(m! {\pi } x)|<1 , so that \lim {m \rightarrow \infty}\left(\lim {n \rightarrow \infty} \cos ^{2 n}(m ! \pi x)\right) =0  for all m>0 by Sandwich Theorem.

Similar Problems and Solutions



ISI MStat 2018 PSA Problem 10
Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

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This is a problem from ISI MStat 2018 PSA Problem 10 based on Dirichlet Function.

Dirichlet Function - ISI MStat Year 2018 PSA Question 10


Let x be a real number. Then \lim {m \rightarrow \infty}\left(\lim {n \rightarrow \infty} \cos ^{2 n}(m ! \pi x)\right)

  • (A) does not exist for any x
  • (B) exists for all x
  • (C) exists if and only if x is irrational
  • (D) exists if and only if x is rational

Key Concepts


Limit

Sandwich Theorem

Check the Answer


Answer: is (B)

ISI MStat 2018 PSA Problem 10

Introduction to Real Analysis by Bertle Sherbert

Try with Hints


Check two cases separately one when x is rational and other is when x is irrational.

If m!x  is an integer, then cos ^{2 n}(m ! \pi x) =1

If x is rational \frac{p}{q}, then, eventually, for large enough m, m! will be divisible by q , so that m!x will be an integer, and we have \lim {m \rightarrow \infty}\left(\lim {n \rightarrow \infty} \cos ^{2 n}(m ! \pi x)\right) =1

If x is irrational, m!x  will never be an integer, and |cos(m! {\pi } x)|<1 , so that \lim {m \rightarrow \infty}\left(\lim {n \rightarrow \infty} \cos ^{2 n}(m ! \pi x)\right) =0  for all m>0 by Sandwich Theorem.

Similar Problems and Solutions



ISI MStat 2018 PSA Problem 10
Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

Subscribe to Cheenta at Youtube


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2 comments on “ISI MStat 2018 PSA Problem 10 | Dirichlet Function”

  1. What I can see that the limit exists for all real x. If x is rational the limit evaluates to 1 and if x is irrational the limit evaluates to 0. Then why do say that the limit doesn't exist for any real x? The reasoning is not quite clear to me. Can you please explain?

    1. Don't worry. It will be (B) i.e limit exists for all real x we have also shown that . I mistakenly put A as option . Between thanks for your valuable reply.

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