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  4. displaystyle lim n arrow 0 (43 n-2-9n+1/82 n-1-9n-1)=

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Q. $\displaystyle\lim _{n \rightarrow 0} \frac{4^{3 n-2}-9^{n+1}}{8^{2 n-1}-9^{n-1}}=$

Limits and Derivatives

Solution:

$ \displaystyle\lim _{n \rightarrow \infty} \frac{4^{3 n-2}-9^{n+1}}{8^{2 n-1}-9^{n-1}}$
$= \displaystyle\lim _{n \rightarrow+\infty} \frac{4^{-2} \cdot 64^{n}-9 \cdot 9^{n}}{8^{-1} \cdot 64^{n}-9^{-1} \cdot 9^{n}} $
$= \displaystyle\lim _{n \rightarrow+\infty} \frac{4^{-2}-9\left(\frac{9}{64}\right)^{n}}{8^{-1}-9^{-1}\left(\frac{9}{64}\right)^{n}} $
$=\frac{4^{-2}-0}{8^{-1}-0}=\frac{1}{2} $
$= \displaystyle\lim _{n \rightarrow-\infty} \frac{4^{-2}\left(\frac{64}{9}\right)^{n}-9}{8^{-1}\left(\frac{64}{9}\right)^{n}-9^{-1}}$
$=\frac{0-9}{0-9^{-1}}=81$
Hence, limit does not exist.