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Lim x→0(1+sin x)^cosecx is equal to Single correct option
lim x → 0 ( 1 + sin x ) cosec x \displaystyle\lim_{x\to0}(1+\sin x)^{\operatorname{cosec}x} x → 0 lim ( 1 + sin x ) cosec x is equal to
A. 1 1 1 B. e 2 e^2 e 2 C. 0 0 0 D. e e e Show answer and solution Answer: (D) e e e
Put
u = sin x → 0 u=\sin x\to0 u = sin x → 0 . Then
( 1 + u ) 1 / u → e (1+u)^{1/u}\to e ( 1 + u ) 1/ u → e , since
cosec x = 1 sin x = 1 u \operatorname{cosec}x=\frac1{\sin x}=\frac1u cosec x = s i n x 1 = u 1 . Trap: seeing the base tend to 1 and concluding the limit is
1 1 1 (the form is
1 ∞ 1^\infty 1 ∞ , which is indeterminate), or squaring
e e e by mistake.
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