Question

If \[ \lim_{x \to 5} \frac{x^k - 5^k}{x - 5} = 500 \], then the value of \( k \), where \( k \in \mathbb{N} \) is

• A. 5
• B. 3
• C. 4
• D. 6

Hint:

Remember \(\lim_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n-1}\). This is also the derivative of \(x^n\) at \(x = a\).

Answer 1

The Correct Option is C

Step 1: Understanding the Question: We have a limit that equals 500. The expression is of the form \(\lim_{x \to a} \frac{x^k - a^k}{x - a}\) with \(a=5\).

Step 2: Key Formula or Approach: Using the standard limit (or definition of derivative): \(\lim_{x \to a} \frac{x^k - a^k}{x - a} = k a^{k-1}\).

Step 3: Detailed Explanation: Apply the formula: \(\lim_{x \to 5} \frac{x^k - 5^k}{x - 5} = k \cdot 5^{k-1}\). Set this equal to 500: \(k \cdot 5^{k-1} = 500\). Test natural numbers \(k\): - \(k=3\): \(3 \cdot 5^{2} = 3 \cdot 25 = 75\) (too small) - \(k=4\): \(4 \cdot 5^{3} = 4 \cdot 125 = 500\) (matches) - \(k=5\): \(5 \cdot 5^{4} = 5 \cdot 625 = 3125\) (too large) Thus \(k = 4\).

Step 4: Final Answer:
\(k = 4\), corresponding to option (C).