Question

The value of \( \lim_{x \to 3} \frac{x^5 - 3^5}{x^8 - 3^8} \) is equal to:

• A. \( \frac{5}{8} \)
• B. \( \frac{5}{64} \)
• C. \( \frac{5}{216} \)
• D. \( \frac{1}{27} \)
• E. \( \frac{1}{63} \)

Hint:

Alternatively, you can use L'Hôpital's Rule for \( 0/0 \) forms: differentiate the numerator and denominator separately. \( \lim \frac{5x^4}{8x^7} = \frac{5(3)^4}{8(3)^7} = \frac{5}{8 \cdot 3^3} \).

Answer 1

The Correct Option is C

Concept: We use the standard limit formula: \( \lim_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n-1} \). To apply this to the given fraction, we divide both the numerator and denominator by \( (x - 3) \).

Step 1: Rearrange the limit expression.
\[ \lim_{x \to 3} \frac{x^5 - 3^5}{x^8 - 3^8} = \lim_{x \to 3} \frac{\frac{x^5 - 3^5}{x - 3}}{\frac{x^8 - 3^8}{x - 3}} \]

Step 2: Apply the limit formula.
Numerator: \( \lim_{x \to 3} \frac{x^5 - 3^5}{x - 3} = 5 \cdot 3^{5-1} = 5 \cdot 3^4 \). Denominator: \( \lim_{x \to 3} \frac{x^8 - 3^8}{x - 3} = 8 \cdot 3^{8-1} = 8 \cdot 3^7 \).

Step 3: Simplify the resulting fraction.
\[ \text{Value} = \frac{5 \cdot 3^4}{8 \cdot 3^7} = \frac{5}{8 \cdot 3^{7-4}} \] \[ \text{Value} = \frac{5}{8 \cdot 3^3} = \frac{5}{8 \cdot 27} \] \[ \text{Value} = \frac{5}{216} \]