Prove that \(\displaystyle{f{{\left({x}\right)}}}={\cos{{\left(\frac{{1}}{{x}}\right)}}}\) is not periodic without

annlanw09y

annlanw09y

Answered question

2022-03-30

Prove that f(x)=cos(1x) is not periodic without using calculus

Answer & Explanation

Jermaine Lam

Jermaine Lam

Beginner2022-03-31Added 11 answers

Assume
cos1x=cos1x+T
Then we must have
1x=±1x+T+2kπ
k=d1xd1x+T2π
But the RHS cannot be an integer for all x, a contradiction.
aznluck4u72x4

aznluck4u72x4

Beginner2022-04-01Added 16 answers

The zeros of f are at the points x such that
1x=π2+kπ
for k a positive integer. Then they are of the form
x=2(2k+1)π<2π
Since the set of zeros is nonempty and bounded, the function cannot be periodic, because if T>0 is a period and x0 is a zero, then also x0+nT would be a zero, for every positive integer n.

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