Limit from some russian book: lmt-0 [(1+mx)^n-(1+nx)^m]/x^2

[mateloco]

limit from some russian book: lmt--0 [(1+mx)^n-(1+nx)^m]/x^2

Hey guys, I'm newbie in calculus, while doing a quick reading of my problem's book i find this interesting limit:

Homework Statement

solve the following limit when x tend to zero,
m and n belong to the set of naturals
$$\frac{(1+mx)^n-(1+nx)^m}{ x^2}$$

Homework Equations

The Attempt at a Solution

the anwser is (mn)(m-n)
i don't know how to solve the problem...

 

[Mark44]

Hey guys, I'm newbie in calculus, while doing a quick reading of my problem's book i find this interesting limit:

Homework Statement

solve the following limit when x tend to zero,
m and n belong to the set of naturals
$$\frac{(1+mx)^n-(1+nx)^m}{ x^2}$$

Homework Equations

The Attempt at a Solution

the anwser is (mn)(m-n)
i don't know how to solve the problem...

The approach I would take would be to expand both binomials using the binomial theorem http://en.wikipedia.org/wiki/Binomial_theorem.

 

[qbert]

or use L'Hospital's rule (twice).
also.. your answer isn't right.

 

[Mark44]

or use L'Hospital's rule (twice).
also.. your answer isn't right.

I like your answer better than mine.

 

[mateloco]

or use L'Hospital's rule (twice).
also.. your answer isn't right.

Exactly!
Before posting, I've tryed L'hopp, and i reached the following answer:
(mn)(m-n)(1/2)
Right?, but looking at the answer's section in the book, i realized that the answer given by the book is:
(mn)(m-n)

Warning! my silly idea:
expanding the binomial, and doing the substitution of all x, almost every term of the function will be zero, it doesn't matter the value of m and n, am i wrong? What do you think about it?
i appreciate all your comments and if you notice any mistakes in my writing (Sintax, grammar, etc.), let me know*(?), English is not my mother tongue, but I want to learn it.

 

[Tedjn]

When you expand with the binomial theorem, in the numerator, what is the coefficient of x? What happens when you divide that by x² as x approaches zero?

EDIT: Also, your idea isn't silly. Most ideas aren't silly but some lead to silly conclusions when you realize that you didn't get anywhere. And some leave you feeling silly since you didn't follow them to completion. But the idea, well, that's just being a creative person.

 

[mateloco]

When you expand with the binomial theorem, in the numerator, what is the coefficient of x? What happens when you divide that by x² as x approaches zero?

EDIT: Also, your idea isn't silly. Most ideas aren't silly but some lead to silly conclusions when you realize that you didn't get anywhere. And some leave you feeling silly since you didn't follow them to completion. But the idea, well, that's just being a creative person.

Very nice comment Tedjn!

 

[gomunkul51]

Alex Spivak ? :)