Math Help - Limit formula derivation

1. Limit formula derivation

Hey guys,

I saw a formula that would be really useful in a proof for the general form of differentiation and I was wondering if it actually held true and if so how it has been derived. I haven't been able to find it anywhere on the internet as google isn't a good tool for formula searching.

Anyways, the formula states,

lim_x->0 (x^m-a^m)/(x^n-a^n) = (m/n)*a^(m-n)

Now I am thinking that you could prove the general form of derivatives from definition by adding and subtracting an x on the bottom of the quotient so the denominator is in the form (x + h) - x which then can be used in this formula to show that f'(x) = n * a^n-1 for f(x) = a^n

Does this work?
And if so does this work for any real number? The only proofs that I've seen using the definition require that the value 'n' has to be an integer, and I haven't seen this formula used to prove a general form equation either.

Does anyone know where this formula comes from or want to guess at where it is derived from?

Thanks,
Mark

2. Re: Limit formula derivation

Originally Posted by MarkJacob
Hey guys,

I saw a formula that would be really useful in a proof for the general form of differentiation and I was wondering if it actually held true and if so how it has been derived. I haven't been able to find it anywhere on the internet as google isn't a good tool for formula searching.

Anyways, the formula states,

lim_x->0 (x^m-a^m)/(x^n-a^n) = (m/n)*a^(m-n)

Now I am thinking that you could prove the general form of derivatives from definition by adding and subtracting an x on the bottom of the quotient so the denominator is in the form (x + h) - x which then can be used in this formula to show that f'(x) = n * a^n-1 for f(x) = a^n

Does this work?
And if so does this work for any real number? The only proofs that I've seen using the definition require that the value 'n' has to be an integer, and I haven't seen this formula used to prove a general form equation either.

Does anyone know where this formula comes from or want to guess at where it is derived from?

Thanks,
Mark
Your statement is not true. Provided \displaystyle \begin{align*} x \neq a \end{align*} we have

\displaystyle \begin{align*} \lim_{x \to 0}\frac{x^m - a^m}{x^n - a^n} &= \frac{0^m - a^m}{0^n - a^n} \\ &= \frac{-a^m}{-a^n} \\ &= \frac{a^m}{a^n} \\ &= a^{m-n} \end{align*}

not \displaystyle \begin{align*} \frac{m}{n}\,a^{m-n} \end{align*}

Are you sure you're not trying to evaluate \displaystyle \begin{align*} \lim_{x \to a}\frac{x^m - a^m}{x^n - a^n} \end{align*}?

3. Re: Limit formula derivation

Oh yeah, it would make more sense if it was x --> a, and it works for this too, I think.
Although with the definition for derivatives, h tending towards zero implies x + h tending towards x, so to prove a general form derivative equation x tending towards a does work, does it not?

4. Re: Limit formula derivation

Originally Posted by MarkJacob
Oh yeah, it would make more sense if it was x --> a, and it works for this too, I think.
Although with the definition for derivatives, h tending towards zero implies x + h tending towards x, so to prove a general form derivative equation x tending towards a does work, does it not?
L'Hospital's Rule works nicely here