Find and prove

$\displaystyle \lim_{n\rightarrow\infty}(\sqrt{n+3} - \sqrt{n})$

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- Mar 15th 2013, 04:21 PMKanwar245Proof of a limit of an infinite sequence
Find and prove

$\displaystyle \lim_{n\rightarrow\infty}(\sqrt{n+3} - \sqrt{n})$ - Mar 15th 2013, 04:21 PMKanwar245Re: Proof of a limit of an infinite sequence
I'm thinking of rationalizing it first, also I can see that the limit approaches 0

- Mar 15th 2013, 04:24 PMHallsofIvyRe: Proof of a limit of an infinite sequence
That sounds good- multiply by $\displaystyle \frac{\sqrt{n+3}+ \sqrt{n}}{\sqrt{n+3}+ \sqrt{n}}$

- Mar 15th 2013, 04:33 PMKanwar245Re: Proof of a limit of an infinite sequence
yes and it becomes

$\displaystyle \frac{3}{(\sqrt{n+3}+\sqrt{n})}$ - Mar 15th 2013, 05:06 PMShakarriRe: Proof of a limit of an infinite sequence
From

$\displaystyle a_n=\frac{3}{\sqrt{n+3}+\sqrt{n}}$

You can show that this is always less than

$\displaystyle b_n=\frac{3}{\sqrt{n}+\sqrt{n}}$

and always greater than

$\displaystyle c_n=0$

Using the sandwich theorem show that a_{n}converges to zero. - Mar 15th 2013, 05:32 PMProve ItRe: Proof of a limit of an infinite sequence
- Mar 15th 2013, 05:44 PMShakarriRe: Proof of a limit of an infinite sequence
Oh yes you could do it that way (Clapping)

- Mar 15th 2013, 06:36 PMKanwar245Re: Proof of a limit of an infinite sequence
right but when proving, for all epsilon greater than 0, what N would I choose so that $\displaystyle |\frac{3}{2\sqrt{n}}| < \epsilon$

If i try to square it, epsilon gets squared too... - Mar 15th 2013, 06:53 PMProve ItRe: Proof of a limit of an infinite sequence
- Mar 15th 2013, 07:18 PMKanwar245Re: Proof of a limit of an infinite sequence
Right so we have this:

$\displaystyle |\frac{3}{\sqrt{n} + \sqrt{n}}| < \epsilon$

$\displaystyle \Rightarrow |\frac{3}{2\sqrt{n}}| < \epsilon$

$\displaystyle \Rightarrow \frac{3}{2\sqrt{n}} < \epsilon$

Now from this point on, if I try to isolate n, at some point I will have to square $\displaystyle \sqrt{n}$ which will yield $\displaystyle {\epsilon}^2$ ... - Mar 15th 2013, 08:59 PMProve ItRe: Proof of a limit of an infinite sequence
- Mar 16th 2013, 07:03 AMKanwar245Re: Proof of a limit of an infinite sequence
- Mar 16th 2013, 03:05 PMProve ItRe: Proof of a limit of an infinite sequence
- Mar 16th 2013, 03:06 PMKanwar245Re: Proof of a limit of an infinite sequence
Okay Perfect!