let-f-0-R-be-a-continuous-function-if-lim-n-0-1-f-x-n-dx-2-then-lim-n-f-nx-

Question Number 206433 by universe last updated on 14/Apr/24

let f : [0, \infty) \to R be a continuous function if

\lim_{n \to \infty} \int_{0}^{1} f (x + n) dx = 2

then \lim_{n \to \infty} f (nx) = ?

Answered by Berbere last updated on 14/Apr/24

\int_{0}^{1} f (x + n) d x \overset{x + n = y}{=} \int_{n}^{1 + n} f (y) d y = 2

i f \lim_{x \to \infty} f (x) e x i s t e l e t a = \lim_{x \to \infty} f (x)

\Rightarrow \forall ϵ > 0 \exists N s u c h t h a t \forall x > N a - ϵ ⩽ f (x) ⩽ a + ϵ

\Rightarrow a - ϵ ⩽ \int_{n}^{n + 1} f (x) ⩽ a + ϵ; \forall n > N

\Rightarrow \lim_{n \to \infty} \int_{n}^{n + 1} f (x) d x = a = 2

t h e n \lim_{n \to \infty} f (n x) = {\begin{cases} f (0) i f x = 0 \\ 2 \forall x > 0 \end{cases}

Commented by aleks041103 last updated on 14/Apr/24

f i s c o n t i n i o u s ⇏ \exists \underset{x \to \infty}{l i m} f (x)

e x a m p l e : f (x) = s i n (x), \underset{x \to \infty}{∄ l i m f (x)}

t h e r e f o r e y o u r s o l u t i o n i s w r o n g

Commented by Berbere last updated on 14/Apr/24

y e s y o u r i g h t i d o n t w h y i s a i d T h a t

Answered by Berbere last updated on 14/Apr/24

f (x) = 2 + c o s (2 π x)

\int_{0}^{1} (2 + c o s (2 π x)) = 2 n o l i m i t e

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