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  4. The value of lim limitsx→∞((x2 -2x +1/x2 -4x +2))x is

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Q. The value of $ \lim\limits_{x\to\infty}\left(\frac{x^{2} -2x +1}{x^{2} -4x +2}\right)^{x} $ is

MHT CETMHT CET 2007

Solution:

$\displaystyle\lim _{x \rightarrow \infty}\left(\frac{x^{2}-2 x+1}{x^{2}-4 x+2}\right)^{x}$
$=\displaystyle\lim _{x \rightarrow \infty}\left(1+\frac{2 x-1}{x^{2}-4 x+2}\right)^{x}$
$=e^{\displaystyle\lim_{x\to\infty}} \left(\frac{x\left(2x-1\right)}{x^{2}-4x+2}\right)=e^{2}$