Changing the limit variable

Question

let r >0 with ;

$f$ : $(0,r) \mapsto \mathbb{R} $

$F$ : $ (\frac{1}{r} , \infty) \mapsto \mathbb{R} $

$F(x)= f(\frac{1}{x}) $

show that :$ \lim_{x \downarrow 0} f(x) = L \Leftrightarrow lim_{x \to \infty} F(x) = L$

so can solve it like that : first : :$ \lim_{x \downarrow 0} f(x) = L \Rightarrow lim_{x \to \infty} F(x) = L$

we know already that , $ \lim_{x \downarrow 0} f(x) $= L let it be then T >0 so that $ T = \frac{1}{x} $ $\lim_{x \to \infty } \frac{1}{x} = \lim_{T \downarrow 0 } T $ then $\lim_{x \to \infty } f(\frac{1}{x}) = \lim_{T \downarrow 0} f(T)= L $

Answer 1

hint

$$X>A>0 \iff 0<\frac{1}{X}<\frac{1}{A}=\eta$$