Q.257. Limit

Question 257.
What is the value of lim ( x → π/2) (sin x)^(tan x) ?
(A)   0,                (B) 1,                (C) e,                (D) π.

Answer 257.
lim (x → π/2) (sinx)^(tanx)
= lim (x → π/2) e^[(tanx) ln (sinx)]
= e^ [lim (x → π/2) (tanx) ln (sinx)] ... (1)

lim (x → π/2) (tanx) ln (sinx)
= lim (x → π/2) [ln(sinx) / cotx]

Using L'Hospital'stheorem,
= lim (x → π/2) [- cotx / cosec^2 x]
= 0

Plugging in ( 1 ),
required limit = e^0 = 1.

=> Answer is (b) 1.

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