Question

The positive integer \(n\), such that \(\lim_{x \to 3} \frac{x^n - 3^n}{x - 3} = 108\)

• A. $3$
• B. $12$
• C. $6$
• D. $9$
• E. $4$

Hint:

Memorize $\frac{x^n - a^n}{x-a} \to n a^{n-1}$ — very frequently used.

Answer 1

Answer: (E)

Concept: \[ \lim_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n-1} \]

Step 1: Apply formula
\[ \lim_{x \to 3} \frac{x^n - 3^n}{x - 3} = n \cdot 3^{n-1} \]

Step 2: Equate with given value
\[ n \cdot 3^{n-1} = 108 \]

Step 3: Try integer values
\[ n=4 \Rightarrow 4 \cdot 3^3 = 4 \cdot 27 = 108 \] Final Conclusion:
Option (E)