squeeze theorem: lim, n->infty, C^(1/n)=1 for 0<C<1

[shakalandro]

I am supposed to be proving that the limit as n -> infinity of C^(1/n) = 1, 0<C<1, using the fact that for C>1, limit n approaches infinity of C^(1/n) = 1. In order to use the squeeze theorem I need a sequence that converges to 1, but is less than C^(1/n), for 0<C<1.

 

[galactus]

Must you use the squeeze theorem?. It isn't really necessary because just looking at the function, 1/n as n->infinity, goes to 0 and C^0=1

 

[shakalandro]

The course is all about proofs, it's easy to find the limit, but I have to prove it. Using the squeeze theorem seems to be the only sensible way here.

 

[Deleted member 4993]

The course is all about proofs, it's easy to find the limit, but I have to prove it. Using the squeeze theorem seems to be the only sensible way here.

What if you use X = 1/C where X > 1.

 

[shakalandro]

What if you use X = 1/C where X > 1.

That does not converge to 1 as n approaches infinity...it converges to 1/C.
Also, 1/C, where 0<C<1, would make x >C^(1/n) not less than C^(1/n), like I want.

 

[galactus]

\(\displaystyle \lim_{n\to \infty}C^{\frac{1}{n}}=1\)

We can prove it this way:

\(\displaystyle \lim_{n\to \infty}C^{\frac{1}{n}}\)

\(\displaystyle =\left(\lim_{n\to \infty}C\right)^{\lim_{n\to \infty}\frac{1}{n}}\)

\(\displaystyle =C^{\left(\lim_{n\to\infty}\frac{1}{n}\right)}\)

\(\displaystyle =C^{\left(\frac{1}{\lim_{n\to \infty}n}\right)}\)

\(\displaystyle =1\)

I don't see anything wrong with that proof.

 

[Deleted member 4993]

What if you use X = 1/C where X > 1.

That does not converge to 1 as n approaches infinity...it converges to 1/C.
Also, 1/C, where 0<C<1, would make x >C^(1/n) not less than C^(1/n), like I want.

Did you try it with a paper & pencil in hand!!!

lim (n? ?) (C[sup:9wcu9lzk]1/n[/sup:9wcu9lzk])

= lim (n? ?) {(1/X)[sup:9wcu9lzk]1/n[/sup:9wcu9lzk]}

= 1/{lim (n? ?)[X][sup:9wcu9lzk]1/n[/sup:9wcu9lzk]}

= 1/1 (...using the fact that for X>1, limit n approaches infinity of X^(1/n) = 1)

 

[shakalandro]

Did you try it with a paper & pencil in hand!!!

Of course I did! Don't be dumb.
You're proof was pretty similar to the proof I eventually worked out.
you must use that C = 1/1/C and then apply the limit of a quotient theorem. The way you did it is close except that the quotient theorem can only be used for limits that we know exist.