Expert: Alon Mandes - 6/4/2009

Question
Hi alon, so I have a few questions about these problems. I'm not sure how to do them with x's in various places.

1.
lim     (1+(11/x))^(x/6)
x->INF

2.
lim       x^(3/x)
x->+INF

3.
lim       x^(3sinx)
x->0+  


could you provide answers as well so I know how to solve these with x's as the base or exponent. thanks.  

Answer
Hi Brooke,

lim     (1+(11/x))^(x/6)  = ?
x->INF

We know that :

Lim    [1+(1/x)]^x = e
x->INF

& we also know that

Lim     [1+(a/x)]^x = e^x
x->INF

Thus,

Lim     [1+(11/x)]^(x/6) =
x->INF
Lim     { [1+(11/x)]^x }^(1/6) = e^(11/6) .
x->INF
-------------------------------------------------------
lim       x^(3/x) =?
x->+INF

In this section I'll explain 2 similar examples :
Lim x^(1/x) & Lim (1/x)^x as x goes to INF .
I claim that :
Lim x^(1/x)=1 & Lim (1/x)^x=0 . Why ?
Note that function x goes to INF & function 1/x goes to 0 . The
only thing that matters here is which function is faster ???
The answer is simple : function 1/x converge to zero much more
faster than the divergence of function x to INF.
---------------------------------------------------------
lim       x^(3sinx) = ?
x->0+  
We claim that the answer is 1 . Because sinx<x & thus sinx converge
more rapidly to zero that function x.


Alon.