Given lim x → 0 f ( 2 x ) − f ( x )...

rjawbreakerca

rjawbreakerca

Answered

2022-07-03

Given lim x 0 f ( 2 x ) f ( x ) x = 0 and lim x 0 f ( x ) = 0 show that lim x 0 f ( x ) x = 0

Answer & Explanation

Melina Richard

Melina Richard

Expert

2022-07-04Added 14 answers

We could simplify all this. Let
ρ ( x ) = sup | y | | x | | f ( 2 y ) f ( y ) y |
We have for n 0
| f ( x ) f ( x 2 n + 1 ) | = | k = 0 n ( f ( x 2 k ) f ( x 2 k + 1 ) ) | k = 0 n ρ ( x ) | x | 2 k + 1 | x | ρ ( x ) ( 1 1 2 n + 1 )
By taking the limit of this inequality when n , it follows that | f ( x ) x | ρ ( x ) x 0 0

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