Question
$\mathop {\lim }\limits_{x \to \pi /4} \frac{{\sqrt 2 \cos x - 1}}{{\cot x - 1}} = $
$ = \mathop {\lim }\limits_{x \to \pi /4} \frac{{\sin x\,(1 + \cot x)}}{{(\sqrt 2 + \sec x)}} = \frac{{\frac{1}{{\sqrt 2 }}(2)}}{{\sqrt 2 + \sqrt 2 }} = \frac{1}{2}.$
वैकल्पिक : $L-$ हॉस्पीटल नियम का प्रयोग करें।
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$f(x)=\left[\begin{array}{ll}{\left[e^{x}\right],} \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,x<0 \\ a e^{x}+[x-1], \,\,\,\,\,\,\,\,\,0 \leq x < 1 \\ b+[\sin (\pi x)], \,\,\,\,\,\,\,\,\,\,\,\,1 \leq x < 2 \\ {\left[e^{-x}\right]-c,} \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,x \geq 2\end{array}\right.$
द्वारा परिभाषित है, जहाँ $a , b , c \in R$ तथा $[ t ]$ महत्तम पूर्णांक फलन है तब निम्न में से कौनसा कथन सत्य है-