Limit Evaluation

Date: 7/7/96 at 16:21:41
From: Anonymous
Subject: Limit Evaluation

I'm in the first year of electronics in Buenos Aires and I have a 
problem with a limit. I have no idea how to begin: 
        
                   Lim     Root x of (x)
                  X -> oo

Limit of root X of x, with X -> infinity.

Date: 7/7/96 at 19:3:28
From: Doctor Anthony
Subject: Re: Limit Evaluation

I believe you are wanting the xth root of x, which is better shown as

    x^(1/x)

Let y = x^(1/x)   
Now, with a variable as a power, it is usual to take logs.

Take logs (base e) of both sides

   ln(y) = (1/x)ln(x)  = ln(x)/x

Now as x -> infinity this becomes inf/inf and we can use l'Hopital's 
rule to see what the ratio is becoming as x approaches its limit.

  LT(as x->infin.)ln(y) = (diff. of top line)/(diff. of bottom line) 
                        = (1/x)/1
                        = 1/x
                         ->0 as x -> infin.

If ln(y) -> 0 then y -> 1  and so Lt(as x-> infin.) x^(1/x) = 1 

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 7/7/96 at 20:58:21
From: Anonymous
Subject: Re: Limit Evaluation

Thank you Mr. Anthony, this is the best WWW of math!!
Hernan Gabriel Zapata. (archie@giga.com.ar)