Thread: proof of 2 limits

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 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.

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

Originally Posted by dopi

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 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.

You are missing something, it should be .

Let 0,+\infty)\mapsto \mathbb{R}" alt="f0,+\infty)\mapsto \mathbb{R}" /> and define 0,+\infty)\mapsto \mathbb{R}" alt="g0,+\infty)\mapsto \mathbb{R}" /> as .


I will use the sequencial approach to limit because this approach is the nicest. Let be a sequence in converging to , i.e. . Then, we know (this is an easy theorem, but try to prove it) . Thus, is a sequence in approaching . Thus, this means . This implies then that . Q.E.D.


The theorem I used was let be a sequence of positive real numbers which is convergent to zero. Then . This theorem goes both ways.

Given means that:
.

Select . Now if .
Or .