Question

Use continuity to evaluate the limit.
lim
x→π
5 sin(x + sin x)

Answer 1

use the fact that sine is continuous
what they want you to write is this
\[\lim_{x\to \pi}5\sin(x+\sin(x))=5\lim_{x\to\pi}\sin(x+\sin(x))\] the first because the property that constants don't effect the limit
then write
\[=5\sin(\lim_{x\to \pi}(x+\sin(x))\] because sine is continuous, you can bring the limit inside the function
more to come....

Answer 2

next step
\[=5\sin(\lim_{x\to \pi}x+\lim_{x\to \pi}\sin(x))\] because the limit of the sum is the sum of the limits

Answer 3

next step
\[=5\sin(\pi+\sin(\lim_{x\to \pi}x))\] because of two reasons, sine is continuous still so you can bring the limit inside, and also because \(\lim_{x\to \pi}x=\pi\) is obvious

Answer 4

next step
\[5\sin(\pi+\sin(\pi))\] for the same reason as before
all this work is really just being silly, as everything is sight is continuous, you just replace all \(x\) you see by \(\pi\)

Answer 5

final step is probably to evaluate
since \(\sin(\pi)=0\) you get
\[5\sin(\pi+0)=5\sin(\pi)=0\]

Answer 6

thank you very much