hi im stuck with this question for a while. its a real analysis topic... i was wondering if anyone can help me by
f is defined on (0,infinity) to R.
i want to prove that lim x->infinity f(x) =L if and only if lim x->0 f(1/x)=L ..the x->0 is a positive so its right limit.
my solution so far
i decided to prove each one seperately and let y =f(x),
but the thing is i was getting confused with the definitions and what to do next when using the definitions.
so for the first equation
lim x->infinity f(x) =L
let epsilon>0, N is in natural numbers, y>=N => ! y- L ! < epsilon
this doesnt seem quite right and i wasnt to sure what to do next.
so for the second limit
lim x->0 f(1/x)=L ..the x->0 is a positive so its right limit.
tilda = 1/epsilon
next 0<x- 0 <1/epsilon
this gives ! f(1/x) - L ! < epsilon
since y= f(x) where the x has to be replaced then ! f(x)- L!< epsilon ...hence the quesiton is proved...
can some one check this and help me correct it please thanks