hi im stuck with this question for a while. its a real analysis topic... i was wondering if anyone can help me by

the question:

f is defined on (0,infinity) to R.

i want to prove thatlim x->infinity f(x) =Lif and only if..the x->0 is a positive so its right limit.lim x->0f(1/x)=L

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

..the x->0 is a positive so its right limit.lim x->0f(1/x)=L

let epsilon>0

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