I need some help on this calculus question. Thank you in advance! . lim (sin 6)c

Question

I need some help on this calculus question. Thank you in advance!
.

Explanation / Answer

lim_(x->0^+) sin^(csc(x))(x) lim_(x->0^+) sin^(csc(x))(x) = lim_(x->0^+) e^(log(sin^(csc(x))(x))): lim_(x->0^+) e^(log(sin^(csc(x))(x))) e^(log(sin^(csc(x))(x))) = exp(csc(x) log(sin(x))): lim_(x->0^+) exp(csc(x) log(sin(x))) lim_(x->0^+) e^(csc(x) log(sin(x))) = e^(lim_(x->0^+) csc(x) log(sin(x))): e^(lim_(x->0^+) csc(x) log(sin(x))) Applying the product rule, write lim_(x->0^+) csc(x) log(sin(x)) as (lim_(x->0^+) csc(x)) (lim_(x->0^+) log(sin(x))): e^(lim_(x->0^+) csc(x) lim_(x->0^+) log(sin(x))) lim_(x->0^+) csc(x) = : e^( lim_(x->0^+) log(sin(x))) lim_(x->0^+) log(sin(x)) = log(lim_(x->0^+) sin(x)): e^( log(lim_(x->0^+) sin(x))) lim_(x->0^+) sin(x) = sin(0) = 0: e^( log(0)) log(0) = -: e^( -) e^( -) = 0

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