Limit Proof

**TexasGirl** · January 6th 2006, 03:06 PM

Hey, I am trying to prove that:

lim(nth root of n)=1 as n approaches infinity and that

lim(nth root of n+1)=1 as n approaces infinity

I do not know how to go about it. Can anybody help??

**Jameson** · January 6th 2006, 04:31 PM

Is using the delta-epsilon definition of a limit or a less rigorous method?

Start with considering any function $f(n)=n^\left(\frac{1}{n}\right)$ . You can do this both graphically and analytically. For the first way, use a calculator to see the trend as n tends to infinity. For the second way, look at the exponential part of the function. What is $\lim_{n\rightarrow\infty}\frac{1}{n}$ ? And what is any number or variable to that answer?

**ThePerfectHacker** · January 6th 2006, 06:18 PM

This happens to be a famous limit
I will prove that $\lim_{n\rightarrow \infty} n^{\frac{1}{n}}$ =1.
Notice that the function $n^{\frac{1}{n}}$ is a monotonic decreasing function, which has a lower bound. Thus, $\lim_{n\rightarrow \infty} n^{\frac{1}{n}}$ has a limit. Call that $L$ thus,
L= $\lim_{n\rightarrow \infty} n^{\frac{1}{n}}$
thus,
$\ln L=\ln (\lim_{n\rightarrow \infty} n^{\frac{1}{n}})$
But because $\ln$ is countinous for that interval,
$\ln L=\lim_{n\rightarrow \infty} \ln (n^{\frac{1}{n}})$
Thus,
$\ln L=\lim_{n\rightarrow \infty} \frac{\ln n}{n}$
But this fits the necessary condition of L'Hopital's Rule,
$\ln L=\lim_{n\rightarrow \infty} \frac{1}{n}=0$
Thus, $\ln L=0$ thus, $L=1$
Finally putting all of this together,
$\lim_{n\rightarrow \infty} n^{\frac{1}{n}}$ =1

**ThePerfectHacker** · January 6th 2006, 06:21 PM

Originally Posted by Jameson

Is using the delta-epsilon definition of a limit or a less rigorous method?

Start with considering any function $f(n)=n^\left(\frac{1}{n}\right)$ . You can do this both graphically and analytically. For the first way, use a calculator to see the trend as n tends to infinity. For the second way, look at the exponential part of the function. What is $\lim_{n\rightarrow\infty}\frac{1}{n}$ ? And what is any number or variable to that answer?

Jameson I believe the rule goes that if the composition of the limits exists then the limit is the functional composition. That means what you said about looking at the exponent part lacks mathematical rigor.

**CaptainBlack** · January 6th 2006, 11:25 PM

Originally Posted by Jameson

For the second way, look at the exponential part of the function. What is $\lim_{n\rightarrow\infty}\frac{1}{n}$ ? And what is any number or variable to that answer?

This is invalid as it pays no attention to the rate of growth of what is
being raised to $1/n$ . Such an argument would apply to

$\lim_{n\rightarrow\infty}f(n)^{\frac{1}{n}}$

for any real function $f(n)$ .

In particular it would appy when $f(n)=n^n$ .

RonL

**CaptainBlack** · January 6th 2006, 11:27 PM

Originally Posted by ThePerfectHacker

Jameson I believe the rule goes that if the composition of the limits exists then the limit is the functional composition. That means what you said about looking at the exponent part lacks mathematical rigor.

- its actualy false (fallacious?), see my other post

RonL

**CaptainBlack** · January 7th 2006, 01:28 AM

Originally Posted by TexasGirl

Hey, I am trying to prove that:

lim(nth root of n)=1 as n approaches infinity and that

lim(nth root of n+1)=1 as n approaces infinity

I do not know how to go about it. Can anybody help??

Now:

$n^{\frac{1}{n}}=e^{\frac{ \ln(n)}{n}}$ ,

and as the exponential function is continuous if the limit exists:

$\lim_{n\rightarrow \infty} n^{\frac{1}{n}}=e^{\lim_{n\rightarrow \infty}\frac{ \ln(n)}{n}}$ ,

so we need only worry about:

$\lim_{n\rightarrow \infty}\frac{ \ln(n)}{n}$ .

Now this is dealt with elsewhere in this thread by ThePerfectHacker, but
an informal short cut is available which tells us what this limit is.

The sort cut is that log functions grow more slowly than any power
of its argument, that is for all $\alpha \epsilon \mathbb{R}$ , there exists a $k$ such that:

$\ln (x) < k.x^{\alpha}$ ,

for all $x$ sufficiently large. Which is sufficient to show that

$\lim_{n\rightarrow \infty}\frac{ \ln(n)}{n^{\beta}}=0$ ,

for any $\beta \epsilon \mathbb{R}$ . Of course this can also be demonstrated
using L'Hopital's rule as is done for $\beta = 1$ elsewhere in this thread.

RonL

**TexasGirl** · January 7th 2006, 07:59 AM

You guys are awesome. I don't know what I would do without you. Thanks a million!

**ThePerfectHacker** · January 7th 2006, 11:27 AM

Originally Posted by CaptainBlack

- its actualy false (fallacious?), see my other post

RonL

What are you talking about CaptainBlack.

**CaptainBlack** · January 7th 2006, 12:24 PM

Originally Posted by ThePerfectHacker

What are you talking about CaptainBlack.

Jameson's statement is a fallacy - you need to consider the rate of
growth of what is raised to $1/n$ not just that the limit
of the exponent is $0$ . Its explained in my response to
Jameson's post.

RonL

**ThePerfectHacker** · January 7th 2006, 02:11 PM

Originally Posted by CaptainBlack

Jameson's statement is a fallacy - you need to consider the rate of
growth of what is raised to $1/n$ not just that the limit
of the exponent is $0$ . Its explained in my response to
Jameson's post.

RonL

I know, I told him that.

For the second thing you cannot just take the natural logarithm of the limit you must first prove it exists. It is a common mistake by just taking the natural logarithm because we are assuming it exits.

**Jameson** · January 8th 2006, 08:18 AM

Originally Posted by CaptainBlack

This is invalid as it pays no attention to the rate of growth of what is
being raised to $1/n$ . Such an argument would apply to

$\lim_{n\rightarrow\infty}f(n)^{\frac{1}{n}}$

for any real function $f(n)$ .

In particular it would appy when $f(n)=n^n$ .

RonL

I don't understand your point. Obviously $lim_{x\rightarrow\infty}n^n=\infty$ I was simply telling her to analytically think about the trend of the exponent. If I made a mistake I apologize, but I really don't see how I was false in showing that as n approaches infinity, the function becomes $n^0$

**Jameson** · January 8th 2006, 08:24 AM

Originally Posted by ThePerfectHacker

Jameson I believe the rule goes that if the composition of the limits exists then the limit is the functional composition. That means what you said about looking at the exponent part lacks mathematical rigor.

Yes it did lack rigor. I like your method better. I make a quick post and I as I said above I was trying to analytically show the trend of the function. I'm still not seeing my fallacy though.

**CaptainBlack** · January 8th 2006, 09:06 AM

Originally Posted by Jameson

I don't understand your point. Obviously $lim_{x\rightarrow\infty}n^n=\infty$ I was simply telling her to analytically think about the trend of the exponent. If I made a mistake I apologize, but I really don't see how I was false in showing that as n approaches infinity, the function becomes $n^0$

Your argument about the exponent if valid would also prove that:

$\lim_{x\rightarrow\infty}(n^n)^{1/n}=1$ ,

as here also the outer exponent goes to zero and as we know anything
raised to the power zero (if non-zero) is unity.

RonL

**ThePerfectHacker** · January 8th 2006, 09:38 AM

Jameson, your problem, informally was that you looked at the exponent you should have also seen the base thus it is of the form $\infty^0$ no conclusion could be drawn.

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