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Question
If \[\lim_{x \to 3} \frac{x^n - 3^n}{x - 3} = 108,\] find the value of n.
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Solution
\[\lim_{x \to 3} \left[ \frac{x^n - 3^n}{x - 3} \right] = 108\]
⇒ x(3)n – 1 = 108
⇒ x(3)n – 1 = 4 × 33
On comparing LHS and RHS, we observe that x is equal to 4.
\[\begin{array}{c|c|c}2 & 108 \\ 2 & 54 \\ 3 & 27 \\ 3 & 9 \\ 3 & 3 \\& 1\end{array}\]
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