How do you find the limit of # [ x (cot^2 x)] / [(csc x) +1]# as x approaches 0?

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Answer 1 · 2016-09-07T08:26:57

#= 1#

#lim_(x to 0) [ x cot^2 x] / [csc x +1]#

#=lim_(x to 0) [ x cot^2 x] / [csc x +1] * (csc x - 1)/(csc x - 1)#

#=lim_(x to 0) [ x cot^2 x (csc x - 1)] / [csc^2 x - 1]#

#=lim_(x to 0) [ x cot^2 x (csc x - 1)] / [cot^2 x ]#

#=lim_(x to 0) x (csc x - 1)#

#=lim_(x to 0) x /(sin x) - x#

and because #lim_(alpha to 0) alpha /(sin alpha) = 1#

#= 1 - 0 = 1#