Question

lf $\underset{x\to 5}{lim}\frac{x^k-5^k}{x-5}=500$, then the positive integral value of k is:

• A. $3$
• B. $4$
• C. $5$
• D. $6$

Answer 1

The correct option is B $4$

$\underset{x\to 5}{lim}\frac{x^k-5^k}{x-5}=500$
Clearly form of the limit is $\frac{0}{0}$, Thus using L-Hospital's rule,
$\underset{x\to 5}{lim}\frac{k{x}^{k-1}}{1}=500$
$\Rightarrow k\times 5^{k-1}=500$
Clearly, $k=4$ is satisfying above equation.