Question

What's the limit of `(x^n-a^n)/(x-a)` as `x` approaches `a` , using the derivate number which is `f'(a)` ?

Answer 1

Please see below.

Explanation:

I'm not sure I understand your question. I don't know what you mean by "using the derivate number which is `f'(a)`"

If I understand it, I think I need to point out that

`lim_(xrarra)(x^n-a^n)/(x-a) = f'(a)` for `f(x) = x^n`.

That is:

For `f(x) = x^n`,

`f'(a) = lim_(xrarra)(f(x) - f(a))/(x-a) = (x^n-a^n)/(x-a)`

We also know, by the power rule for derivatives,

That for `f(x) = x^n`, we haved `f'(x) = nx^(n-1)` and `f'(a) = na^(n-1)`.

We can conclude that

`lim_(xrarra)(x^n-a^n)/(x-a) = na^(n-1)`