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  4. displaystyle lim x arrow 0[ min (y2-4 y+11) ( sin x/x)] (where [⋅] denotes the greatest integer function) is

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Q. $\displaystyle\lim _{x \rightarrow 0}\left[\min \left(y^{2}-4 y+11\right) \frac{\sin x}{x}\right]$ (where $[\cdot]$ denotes the greatest integer function) is

Limits and Derivatives

Solution:

$\min \left(y^{2}-4 y+11\right)=\min \left[(y-2)^{2}+7\right]=7$
or $L =\displaystyle \lim _{x \rightarrow 0}\left[\min \left(y^{2}-4 y+11\right) \frac{\sin x}{x}\right]$
$=\displaystyle \lim _{x \rightarrow 0}\left[\frac{7 \sin x}{x}\right]$
$=[$ a value slightly lesser than 7$]=6 .$
$(|\sin x|<|x|$, when $x \rightarrow 0)$