Question

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

• A. $3$
• B. $4$
• C. $5$
• D. $6$

Answer 1

Answer: (B)

$4$

${lim}_{x\to 5}\frac{x^k-5^k}{x-5}=500$

Substituting the x by 5.

As we can see this is $\frac{0}{0}$ indeterminate form, hence we use L'Hospital's Rule

$\Rightarrow {lim}_{x\to 5}\frac{k{\left(x\right)}^{k-1}-0}{1-5}=500$

Substituting $x=5$

$\Rightarrow k{\left(5\right)}^{k-1}=500$

$\Rightarrow k{\left(5\right)}^{k-1}=4\times 5^{4-1}$

$\Rightarrow k=4$

Hence B is the correct option.