Question

If $\underset{x\to 3}{lim}\frac{x^n-3^n}{x-3}=108$ and $n\in N$, find $n$

Answer 1

$\underset{x\to 3}{lim}\frac{x^n-3^n}{x-3}$

Use L_Hospitals rules.

$\frac{n{x}^{n-1}}{\left(1\right)}=n\times (3{)}^{n-1}=108$

$n(3{)}^{n-1}=108$

$n=4$.