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Square Roots and Limits

```Date: 03/12/2003 at 21:39:19
From: Jonathan Coveney
Subject: Square Roots

Given a value of x, find the value of the expression sqrt(x sqrt(x
sqrt(x... and so on, and explain why.
```

```
Date: 03/13/2003 at 03:41:32
From: Doctor Ian
Subject: Re: Square Roots

Hi Jonathan,

The easiest way to deal with this is to make use of the fact that

sqrt(x)

and

x^(1/2)

are just two ways of saying the same thing. If that doesn't seem
familiar to you, take a look at

Properties of Exponents
http://mathforum.org/library/drmath/view/57293.html

So,

_______
|     __          1/2 1/2
\| x \| x  = (x * x   )

3/2 1/2
= (x   )

(3/2 * 1/2)
= x

3/4
= x

and
____________
|     _______
|    |     __         3/4 1/2
\| x \| x \| x = (x * x   )

7/4 1/2
= (x   )

(7/4 * 1/2)
= x

7/8
= x

At each step, the exponent changes from

(k-1)/k

to

((k + (k-1))/k)/2 = (2k - 1)/(2k)

More generally, for n square roots, the exponent is

(2^n - 1)/(2^n)

So you're right that as n approaches infinity, the exponent approaches
1, and the value of the expression approaches x.

However, note that the denominator of the exponent will always be a
power of 2, so x^(19/20) isn't really part of the sequence.

Now for a bit of terminology. The way that we express this situation
is to say

Limit    (2^n - 1)/(2^n)  = 1
n->inf

"The limit, as n goes to infinity, of [the expression] is
equal to 1."

This is somewhat subtle.  Note that we _don't_ say that the expression
itself,

(2^n - 1)/(2^n)

is ever actually equal to 1, because that's never the case. And we
_don't_ say that n ever _becomes_ 'infinity', because infinity isn't a
number, so we can't use it in arithmetical expressions.

We're saying that we can make the value of the expression as close to
1 as we wish, by choosing a suitably high value for n.

Does this help?

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 03/13/2003 at 08:29:19
From: Jonathan Coveney
Subject: Thank you (Square Roots)

Thanks, I wondered what limits were, exactly. But now it makes plain
sense.
```
Associated Topics:
High School Calculus
High School Polynomials
High School Square & Cube Roots

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