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Q.
If $\displaystyle\lim_{x \to 5} \frac{x^k - 5^k}{x - 5} = 500$ then k is equal to :
Limits and Derivatives
Solution:
Let $\displaystyle\lim_{x \to 5} \frac{x^k - 5^k}{x - 5} = 500$
By using $\displaystyle\lim_{x \to a} \frac{x^n - a^n}{x - a} = n.a^{n -1} $ we have $k.5^{k -1} = 500$
Now, put $k = 4$, we get
$4.5^{4 -1} = 500$
$\Rightarrow \:\: 4.5^3 = 500$ which is ture.
$\therefore \:\:\: k = 4$