Use continuity to evaluate the limit.

Question

Use continuity to evaluate the limit.

$$\lim_{x\to \pi} 8\sin(x+\sin x)$$

Don't really understand this. My trig func, knowledge is low but an explanation to look back to always helps me move forward.

Answer 1

"Using continuity" means use the fact that if $f$ is continuous, then $f(a) = \lim_{x \to a} f(x)$. In your case $f(x) = 8\sin(x+\sin(x))$ is continuous, so $$\lim_{x \to \pi} f(x) = f(\pi) = 8\sin(\pi + \sin(\pi)) = 8\sin(\pi) = 0.$$

Answer 2

With continuity, the value of the limit is equal to the expression evaluated at the limiting value of $x$. (I.e., you get the correct limit by plugging in the limiting value of $x$.)

Answer 3

The phrase "$f$ is continuous at $a$" really means

$$\lim_{x\to a} f( \mbox{ junk }) = f(\lim_{x\to a} \mbox{ junk }).$$

Well, not quite. This is what "$f$ is continuous at whatever $\lim_{x\to a}$ junk is."

So the start to your exercise is to write"

$$\lim_{x \to \pi} 8 \sin(x+\sin x) = 8\sin(\lim_{x\to \pi}(x+\sin x)).$$