lets-f-0-R-x-y-f-x-f-y-g-0-R-if-x-0-f-x-g-x-f-2x-lim-x-f-x-L-L-is-finite-does-lim-x-f-x-g-x-0-

Question Number 4297 by 123456 last updated on 07/Jan/16

lets

f : [0, + \infty) \to R, \forall x ⩾ y \Rightarrow f (x) ⩾ f (y)

g : [0, + \infty) \to R

if

\forall x \in [0, + \infty), f (x) ⩽ g (x) ⩽ f (2 x)

\lim_{x \to + \infty} f (x) = L, L is finite

does

\lim_{x \to + \infty} f (x) - g (x) = 0 ?

Commented by Yozzii last updated on 08/Jan/16

\lim_{x \to \infty} f (2 x) = \lim_{0.5 u \to \infty} f (u) = \lim_{u \to \infty} f (u) = L

S i n c e f (x) ⩽ g (x) ⩽ f (2 x) \forall x \in [0, \infty)

\Rightarrow \lim_{x \to \infty} f (x) ⩽ \lim_{x \to \infty} g (x) ⩽ \lim_{x \to \infty} f (2 x) .

\lim_{x \to \infty} f (x) = \lim_{x \to \infty} f (2 x) = L \Rightarrow \lim_{x \to \infty} g (x) = L

b y t h e s q u e e z e t h e o r e m .

\lim_{x \to \infty} {f (x) - g (x)} = \lim_{x \to \infty} f (x) - \lim_{x \to \infty} g (x)

= L - L

= 0 .

(U n d e r c o n s t r u c t i o n)

Commented by prakash jain last updated on 08/Jan/16

f (x) = x

g (x) = 1.5 x

x ⩾ 0, f (x) ⩽ g (x) ⩽ f (2 x)

f (x) - g (x) = - . 5 x

\lim_{x \to + \infty} f (x) - g (x) \neq 0

Commented by Yozzii last updated on 08/Jan/16

\lim_{x \to \infty} f (x) = \lim_{x \to \infty} x = \infty (n o t f i n i t e)

I^{'} m s e a r c h i n g f o r a c o u n t e r e x a m p l e

a s w e l l .

Commented by prakash jain last updated on 08/Jan/16

oh . I did not pay attention .

Commented by prakash jain last updated on 08/Jan/16

You are correct .

\lim_{x \to \infty} f (x) - g (x) = 0