How do you use continuity to evaluate the limit sin(x+sinx)?

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Answer 1 · 2015-09-26T18:45:26

See explanation

Long version:

Sine is continuous so #lim_(xrarra)sinx = sina#

#x# is continuous so, #lim_(xrarra)x = a#

The sum of continuous functions is continuous, so
#lim_(xrarra)(x+sinx) = lim_(xrarra)x+ lim_(xrarra)sinx = a+sina#

The composition of continuous functions is continuous, so #lim_(xrarra)(sin(x+sinx)) = sin(lim_(xrarra) (x+sinx)) = sin(a+sina)#

Short version:

Because of various facts about continuity, we evaluate the limit of #sin(x+sinx)# by substitution.