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  4. displaystyle limx → 0[(sin[x-3]/[x-3])], where [ . ] denotes greatest integer function is

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Q. $\displaystyle\lim_{x \to 0}\left[\frac{sin\left[x-3\right]}{\left[x-3\right]}\right]$, where [ . ] denotes greatest integer function is

Limits and Derivatives

Solution:

$\displaystyle\lim_{x \to 0}\left[\frac{sin\left[x-3\right]}{\left[x-3\right]}\right]$
For $x \to 0^{+}, \left[x -3\right] = -3$
$\therefore \frac{sin\left[x-3\right]}{\left[x-3\right]} = \frac{sin\left(-3\right)}{-3} = \frac{sin\,3}{-3} \in \left(0, 1\right)$
$\therefore \displaystyle\lim_{x \to0^{+}} \frac{sin\left[x-3\right]}{\left[x-3\right]} = 0$
For $x \to 0^{-}, \left[x -3\right] = -4$
$\therefore \frac{sin\left[x-3\right]}{\left[x-3\right]} = \frac{sin\,4}{4}$ lies in $\left(-1, 0\right)$
$\therefore \displaystyle\lim_{x \to 0^{-}} \frac{sin\left[x-3\right]}{\left[x-3\right]} = -1 \therefore $ Limit does not exist.