Question

find the limit and determine if the function is countinuous at the point being approached. limit-->3pi sin(4x-sin4x)

Answer 1

\[\lim_{x\to3\pi}\sin(4x-\sin4x)=\sin\left(\lim_{x\to3\pi}(4x-\sin4x)\right)\]
because sine is continuous.

Answer 2

which equals
\[\sin\left(4\lim_{x\to3\pi}x-\lim_{x\to3\pi}\sin4x\right)\]
which equals
\[\sin\left(4\lim_{x\to3\pi}x-\sin\left(\lim_{x\to3\pi}4x\right)\right)\]
Basically, you're using the continuity of the sine function to say that the limit of the outermost function is the same as the function of the limit of the inner function(s).