A question about limits

TeoLeo · 2014-10-23 00:56:52

Ok guys, I can't work out this sequence limit:

lim sin( π√(4n^2+n) ) = ?
n⇒∞

I'm sure it's not really that hard, but...help, please!!

bobbym · 2014-10-23 03:20:29

Hi;

I would say that limit does not exist.

Olinguito · 2014-10-23 03:20:59

This is a tricky one. Consider

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Hence

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Last edited by Olinguito (2014-10-23 03:22:28)

bobbym · 2014-10-23 03:56:33

Hi;

Are you sure you are allowed to take the limit inside the sin() like that?

Olinguito · 2014-10-23 04:11:24

For continuous functions f, yes:

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In fact this is part of the definition of continuousness!

bobbym · 2014-10-23 04:17:09

Hi;

But this is not a continuous function. It says a sequence limit, implying a discrete function. I am not trying to be a nit, the experts agree with your answer. I am just trying to see if I can understand the question like you do. Thanks.

Olinguito · 2014-10-23 04:32:11

bobbym wrote:

But this is not a continuous function.

Of course the it is! It’s the sine function.

Wikipedia wrote:

For any continuous function f, if
then
.

http://en.wikipedia.org/wiki/Limit_of_a … Properties

Last edited by Olinguito (2014-10-23 04:35:31)

bobbym · 2014-10-23 04:33:48

Hi;

Okay, thanks for the help.

Olinguito · 2014-10-23 04:57:30

Here’s what I did on Excel:

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bobbym · 2014-10-23 05:01:46

Hi;

Yes, that is true but notice that you also made the decision to treat n as an integer. If you did not then the limit would not exist.

Olinguito · 2014-10-23 05:10:40

bobbym wrote:

Hi;
Yes, that is true but notice that you also made the decision to treat n as an integer. If you did not then the limit would not exist.

I agree.

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definitely does not exist. But

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does – since in this case we ignore all values that are not the image of integer n.

bobbym · 2014-10-23 05:11:41

So why choose n as an integer?

Olinguito · 2014-10-23 05:14:50

Why not? We’re dealing with sequences here, aren’t we?

Last edited by Olinguito (2014-10-23 05:18:26)

bobbym · 2014-10-23 05:21:14

That is the answer. I did not see that and assumed a function limit.

Olinguito · 2014-10-23 05:28:12

Let’s see what the OP says.

TeoLeo · 2014-10-23 07:28:12

Hi, Olinguito.
I don't really understand the first part with "Consider: √(4n^2+n) - 2n" .

What's with that 2n?

understand everything else after that...so is "-2n" added there just randomly to help us? Or why did you chooose it?

Last edited by TeoLeo (2014-10-23 07:45:06)

Olinguito · 2014-10-23 20:36:29

Notice that

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That’s pretty useless to us. What we need is more than that: we need to show

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In order to do that we show

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Last edited by Olinguito (2014-10-23 20:42:43)

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A question about limits

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