Question

Solve $\underset{x\to a}{lim}\frac{x^n-a^n}{x-a}=n{a}^{n-1}$ where $n\in N;a>0$

Answer 1

$\underset{x\to a}{lim}\frac{x^n-a^n}{x-a}$
Let $f\left(x\right)=x^n$ and $f^{\prime }\left(x\right)=n{x}^{n-1}$
so, $f\left(a\right)=a^n$
Now, By first principle,
$f^{\prime }\left(a\right)=\underset{n\to 0}{lim}\frac{f(n+a)-f\left(a\right)}{n}$
let h = x - a
as $h\to 0,x\to a$
$\Rightarrow f^{\prime }\left(a\right)=\underset{h\to 0}{lim}\frac{f\left(x\right)-f\left(a\right)}{x-a}$
$\Rightarrow n{a}^{n+}=\underset{x\to a}{lim}\frac{x^n-a^n}{x-a}$