Lim [(x+h)²ln(x+h) - x²ln(x)] / h

h→0

Of course this is finding the derivative of the obvious function, but I would like to know if anyone can evluate this WITHOUT using L'Hospital.

Thanks.

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- February 15th 2008, 12:13 PMspammanon[SOLVED] Difficult limit
Lim [(x+h)²ln(x+h) - x²ln(x)] / h

h→0

Of course this is finding the derivative of the obvious function, but I would like to know if anyone can evluate this WITHOUT using L'Hospital.

Thanks. - February 15th 2008, 12:54 PMThePerfectHacker
- February 15th 2008, 12:58 PMspammanonThanks, but...
Hello,

As I clearly stated, I know this is the derivative of a function. What I am asking is how to evaluate the limit. That is, rearrange the terms or use other algebraic manipulations so that the limit is clearly seen. So, anyone else?

How do you evaluate the limit of the expression without using L'Hospital's rule.

Thanks. - February 15th 2008, 01:58 PMThePerfectHacker

.

Thus, we get,

.

We claim that,

.

Note that,

.

Now,

[tex]\frac{1}{x} = \int_x^{x+h} \frac

Thus, .

I do not have time to finish this. - February 15th 2008, 02:13 PMSoroban
Hello, spammanon!

An interesting challenge . . .

Quote:

Given: . , find

. . using the defintion of the derivative: .

The numerator is: .

. . .

. . .

. . .

. . .

. . .

. . .

. . .

Divide by

. . .

. . .

. . .

Take the limit: .

. . .

. . .

. . .

Therefore: . . . . ta-*DAA!*

- February 15th 2008, 04:30 PMspammanonThanks to you both
Well done. I had duplicated your work precisely, but the insight about (or recognition of) e was the missing link. Thanks, and I hope you enjoyed it!

- February 15th 2008, 04:35 PMbobak
- February 15th 2008, 04:37 PMKrizalid
He actually said so above with a "Thanks to you both".

:D