Let be two sequences verifying that and Compute as

Note: you can't use the MVT.

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- Aug 24th 2009, 08:21 PMKrizalidLimit of the quotient of two sequences
Let be two sequences verifying that and Compute as

__Note__: you can't use the MVT. - Aug 24th 2009, 08:50 PMluobo
- Aug 24th 2009, 08:58 PMKrizalid
You said it, easily, and yes, that's the answer.

But that's quite boring!! That's why I posted the problem here! Try it without those things. - Aug 24th 2009, 09:06 PMluobo
- Aug 24th 2009, 11:36 PMsimplependulum
when ,

can we say the integral and

thus the limit is ?

In fact , we are able to evaluate those integrals no matter what the up. and low. limit are . (Happy) - Aug 24th 2009, 11:52 PMred_dog
Let .

f is strictly increasing. Then

(1)

Let .

g is strictly increasing. Then

(2)

Multiply inequalities (1) and (2):

Now apply limit as and we get - Aug 25th 2009, 05:25 AMluobo
- Aug 25th 2009, 06:57 AMKrizalid
That's nice, different solutions.

I still have a different one, I'll post it when I get back home. - Aug 25th 2009, 09:31 AMWilmer
I did that in my head...took 2.3333... seconds;

but I forgot how (Crying) - Aug 25th 2009, 04:00 PMKrizalid
As promised:

Put and suppose that is continuous at We have so let's prove that as

Note that is and besides given we have

Hence

Finally, on put and we'll get in the same fashion for thus - Aug 25th 2009, 05:12 PMluobo
Define

- Aug 25th 2009, 06:27 PMluobo
Summary, the following methods have been used:

(1) Mean-value Theorem (not allowed here);

(2) L'Hospital's Rule;

(3) Squeeze Theorem;

(4) Definition of Limit;

(5) Definition of Derivative;

(6) .....?