$lim_{x \to 0} [min (y^{2} - 4 y + 11) \cdot \frac{\sin x}{x}]$ (where $[\cdot]$ denotes the greatest integer function) is

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Detailed Solution

$lim_{x \to 0} [min (y^{2} - 4 y + 11) \cdot \frac{\sin x}{x}]$
$min (y^{2} - 4 y + 11) = [min (y - 2)^{2} + 7] = 7$

$\Rightarrow L = lim_{x \to 0} [min (y^{2} - 4 y + 11) \frac{\sin x}{x}]$

$= lim_{x \to 0} [\frac{7 \sin x}{x}] (∵ \frac{\sin x}{x} < 1) At \begin{aligned} x = 0 LHL \end{aligned} lim_{x \to 0^{-}} [7 \cdot \frac{\sin x}{x}] = [7 lim_{x \to 0^{-}} \frac{\sin x}{x}] = 6 At \begin{aligned} x = 0 RHL \end{aligned} lim_{x \to 0^{+}} [7 \cdot \frac{\sin x}{x}] = 6 L H L = R H L \Rightarrow lim_{x \to 0} [\frac{7 \sin x}{x}] = 6$