A) 3
B) 4
C) 5
D) 6
Correct Answer: B
Solution :
We know that, \[\underset{x\to a}{\mathop{\lim }}\,\,\frac{{{x}^{n}}-{{a}^{n}}}{x-a}=n\,\,{{a}^{n-1}}\] \\[\underset{x\to 5}{\mathop{\lim }}\,\frac{{{x}^{k}}-{{5}^{k}}}{x-5}=k{{(5)}^{k-1}}\]; But given, \[\underset{x\to 5}{\mathop{\lim }}\,\,\frac{{{x}^{k}}-{{5}^{k}}}{x-5}=500\], \ \[k{{(5)}^{k-1}}=500\]; \[k\,{{(5)}^{k-1}}=4\,{{(5)}^{4-1}},\,\,\,\,\therefore \,\,\,k=4\].
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