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

1 Answer
Sep 26, 2015

See explanation

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.