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

Sep 26, 2015

See explanation

#### Explanation:

Long version:

Sine is continuous so ${\lim}_{x \rightarrow a} \sin x = \sin a$

$x$ is continuous so, ${\lim}_{x \rightarrow a} x = a$

The sum of continuous functions is continuous, so
${\lim}_{x \rightarrow a} \left(x + \sin x\right) = {\lim}_{x \rightarrow a} x + {\lim}_{x \rightarrow a} \sin x = a + \sin a$

The composition of continuous functions is continuous, so ${\lim}_{x \rightarrow a} \left(\sin \left(x + \sin x\right)\right) = \sin \left({\lim}_{x \rightarrow a} \left(x + \sin x\right)\right) = \sin \left(a + \sin a\right)$

Short version:

Because of various facts about continuity, we evaluate the limit of $\sin \left(x + \sin x\right)$ by substitution.