Question

If $\underset{x\to 3}{\lim }\frac{x^n-3^n}{x-3}=108,$ find the value of n.

Answer 1

$\underset{x\to 3}{\lim }\left[\frac{x^n-3^n}{x-3}\right]=108$

⇒ x(3)^{n – 1} = 108
⇒ x(3)^{n – 1} = 4 × 3³

On comparing LHS and RHS, we observe that x is equal to 4.

$\begin{array}{cc}2& 108\\ 2& 54\\ 3& 27\\ 3& 9\\ 3& 3\\ & 1\end{array}$