Correct Answer - Option 2 : 4
Concept:
\(\rm \displaystyle \lim_{x \rightarrow a} \frac{x^n -a^n}{x-a}=na^{n-1}\)
Calculation:
As we know that, \(\rm \displaystyle \lim_{x \rightarrow a} \frac{x^n -a^n}{x-a}=na^{n-1}\)
∴ \(\rm \displaystyle \lim_{x \rightarrow 4} \frac{x^n -4^n}{x-4}=n4^{n-1}\)
Given: \(\rm \displaystyle \lim_{x \rightarrow 4} \frac{x^n -4^n}{x-4}=256\)
So, n × 4n-1 = 256
⇒ n × 4n-1 = 4 × 64
⇒ 4n-1 = 64
⇒ 4n-1 = 43
⇒ n - 1 = 3
∴ n = 4
