let f:[0,∞)→R be a continuous function if lim_(n→∞ ) ∫_0 ^1 f(x+n)dx = 2 then lim


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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
	$∫_0 ^1 f(x+n)dx=^(x+n=y) ∫_n ^(1+n) f(y)dy=2 if lim_(x→∞) f(x) existe let a=lim_(x→∞) f(x) ⇒∀ε>0 ∃N such that ∀ x>N a−ε≤f(x)≤a+ε ⇒a−ε≤∫_n ^(n+1) f(x)≤a+ε;∀n>N ⇒lim_(n→∞) ∫_n ^(n+1) f(x)dx=a=2 then lim_(n→∞) f(nx)= { ((f(0) if x=0)),((2 ∀x>0)) :}$
	$\int_{0}^{1} f (x + n) d x \overset{x + n = y}{=} \int_{n}^{1 + n} f (y) d y = 2$ $i f \lim_{x \to \infty} f (x) e x i s t e l e t a = \lim_{x \to \infty} f (x)$ $\Rightarrow \forall ϵ > 0 \exists N s u c h t h a t \forall x > N a - ϵ ⩽ f (x) ⩽ a + ϵ$ $\Rightarrow a - ϵ ⩽ \int_{n}^{n + 1} f (x) ⩽ a + ϵ; \forall n > N$ $\Rightarrow \lim_{n \to \infty} \int_{n}^{n + 1} f (x) d x = a = 2$ $t h e n \lim_{n \to \infty} f (n x) = {\begin{cases} f (0) i f x = 0 \\ 2 \forall x > 0 \end{cases}$
	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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