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  4. If L = displaystyle lim x arrow ∞( x - x 2 log e (1+(1/ x ))), then the value of 8 L is

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Q. If $L =\displaystyle\lim _{ x \rightarrow \infty}\left( x - x ^{2} \log _{ e }\left(1+\frac{1}{ x }\right)\right)$, then the value of $8 L$ is

Limits and Derivatives

Solution:

Let $x=1 / y$
$\Rightarrow \displaystyle\lim _{x \rightarrow \infty}\left(x-x^{2} \log _{e}\left(1+\frac{1}{x}\right)\right)$
$=\displaystyle\lim _{y \rightarrow 0}\left(\frac{1}{y}-\frac{\log _{e}(1+y)}{y^{2}}\right)$
$=\displaystyle\lim _{y \rightarrow 0}\left(\frac{y-\log _{e}(1+y)}{y^{2}}\right)$
$=\displaystyle\lim _{y \rightarrow 0}\left(\frac{y-\left(y-\frac{y^{2}}{2}\right)}{y^{2}}\right)=1 / 2$