This is a problem from ISI MStat 2018 PSA Problem 10 based on Dirichlet Function.
Let be a real number. Then
Limit
Sandwich Theorem
Answer: is (B)
ISI MStat 2018 PSA Problem 10
Introduction to Real Analysis by Bertle Sherbert
Check two cases separately one when x is rational and other is when x is irrational.
If is an integer, then
If x is rational , then, eventually, for large enough m, m! will be divisible by q , so that
will be an integer, and we have
If x is irrational, will never be an integer, and
, so that
for all m>0 by Sandwich Theorem.
This is a problem from ISI MStat 2018 PSA Problem 10 based on Dirichlet Function.
Let be a real number. Then
Limit
Sandwich Theorem
Answer: is (B)
ISI MStat 2018 PSA Problem 10
Introduction to Real Analysis by Bertle Sherbert
Check two cases separately one when x is rational and other is when x is irrational.
If is an integer, then
If x is rational , then, eventually, for large enough m, m! will be divisible by q , so that
will be an integer, and we have
If x is irrational, will never be an integer, and
, so that
for all m>0 by Sandwich Theorem.
What I can see that the limit exists for all real
If
is rational the limit evaluates to
and if
is irrational the limit evaluates to
Then why do say that the limit doesn't exist for any real
? The reasoning is not quite clear to me. Can you please explain?
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.