lim x→π/2 (cosec x)^tanx

1.	lim x→π/2 (cosec x)^tanx
Answer» \(\lim\limits_{x \to \frac \pi2} (cosec \, x)^{tan \, x}\) \((1^\infty\, type)\) \(= Exp \left\{\lim\limits_{x\to \frac\pi2} (cosec\, x-1) tan\,x\right\}\) \(= Exp \left\{\lim\limits_{x\to \frac \pi2} \frac {1 - sin \,x}{cos \,x}\right\}\) \(\left(\frac 00 - case\right)\) \(= Exp \left\{\lim\limits_{x\to \frac \pi2} \frac {-cos \,x}{-sin\,x}\right\}\) (By using D.L.H. Rule) \(= Exp \left\{\lim\limits_{x\to \frac \pi2} cot \, x\right\}\) \(= e^0 = 1\) \((\because cot \frac \pi 2 = 0)\) Hence, \(\lim\limits_{x\to \frac \pi 2} (cosec \, x)^{tan\, x} = 1\).

Answer»

\(\lim\limits_{x \to \frac \pi2} (cosec \, x)^{tan \, x}\) \((1^\infty\, type)\)

\(= Exp \left\{\lim\limits_{x\to \frac\pi2} (cosec\, x-1) tan\,x\right\}\)

\(= Exp \left\{\lim\limits_{x\to \frac \pi2} \frac {1 - sin \,x}{cos \,x}\right\}\) \(\left(\frac 00 - case\right)\)

\(= Exp \left\{\lim\limits_{x\to \frac \pi2} \frac {-cos \,x}{-sin\,x}\right\}\) (By using D.L.H. Rule)

\(= Exp \left\{\lim\limits_{x\to \frac \pi2} cot \, x\right\}\)

\(= e^0 = 1\) \((\because cot \frac \pi 2 = 0)\)

Hence, \(\lim\limits_{x\to \frac \pi 2} (cosec \, x)^{tan\, x} = 1\).

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