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

In summary, the conversation discusses a limit problem involving two binomials and the use of L'Hospital's rule. The correct answer to the problem is given as (mn)(m-n), but one person mistakenly thinks it is (mn)(m-n)(1/2). The idea of expanding the binomials is also mentioned, but it is pointed out that when dividing by x^2 as x approaches zero, most terms will become zero. The conversation ends with a compliment and a playful question.
  • #1
mateloco
3
0
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
[tex]\frac{(1+mx)^n-(1+nx)^m}{ x^2}[/tex]


Homework Equations



The Attempt at a Solution


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








 
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  • #2


mateloco said:
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
[tex]\frac{(1+mx)^n-(1+nx)^m}{ x^2}[/tex]


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.
 
  • #3


or use L'Hospital's rule (twice).
also.. your answer isn't right.
 
  • #4


qbert said:
or use L'Hospital's rule (twice).
also.. your answer isn't right.
I like your answer better than mine.
 
  • #5


qbert said:
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.
 
  • #6


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.
 
Last edited:
  • #7


Tedjn said:
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! :cool:
 
  • #8


Alex Spivak ? :)
 

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

1. What is the limit of the function (1+mx)^n-(1+nx)^m/x^2 as x approaches 0?

The limit of the given function as x approaches 0 is equal to 0.

2. How do you find the limit of a function at a specific point?

To find the limit of a function at a specific point, you can use the substitution method by plugging in the desired value for the variable in the function. Alternatively, you can use algebraic manipulation to simplify the function and then evaluate the limit.

3. Can the limit of a function change depending on the direction of approach?

Yes, the limit of a function can change depending on the direction of approach. This is known as a one-sided limit and can be evaluated separately from the overall limit of the function.

4. What is the difference between the limit and the value of a function at a point?

The limit of a function at a point is the value that the function approaches as the input approaches that specific point. The value of a function at a point, on the other hand, is the actual output of the function when the input is that specific point.

5. How do you determine if a limit exists for a function?

A limit exists for a function if the left-hand and right-hand limits are equal. This can be checked by evaluating the limit from both sides of the point in question. If they are equal, then the limit exists. If they are not equal, then the limit does not exist.

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