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  4. displaystyle limx→ π (|x + π|/ sin x)

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Q. $\displaystyle\lim_{x\to \pi } \frac{|x + \pi|}{\sin \,x}$

Limits and Derivatives

Solution:

$\displaystyle\lim_{x\to - \pi } \frac{|x + \pi|}{\sin \,x} = \displaystyle\lim_{t \to 0} \frac{|t|}{\sin (t - \pi)} $
Put $x + \pi = t = \displaystyle\lim_{t \to 0} \frac{|t|}{- \sin \, t}$
LHL imit = $\displaystyle\lim_{t \to 0^-} \frac{-t}{-\sin \, t} = 1$ ,
RHL imit = $\displaystyle\lim_{t \to 0^+} \frac{t}{-\sin \, t} = -1$.
$\therefore $ Limit does not exist.