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Q.
Find the positive integer $n$ so that $\displaystyle \lim_{x \to 3}$ $\frac{x^{n}-3^{n}}{x-3}=108$.
Limits and Derivatives
Solution:
We have, $\displaystyle \lim_{x \to 3}$ $\frac{x^{n}-3^{n}}{x-3}$
$=n(3)^{n-1}$
Therefore, $n(3)^{n-1}=108$
$=4(27)=4(3)^{4-1}$
On comparing, we get $n = 4$