Question

use continuity to evaluate the limit. $lim_{x
ightarrowpi}4sin(x + sin x)$

Explanation:

Step1: Recall continuity property

If \(y = f(x)\) is continuous at \(x = a\), then \(\lim_{x
ightarrow a}f(x)=f(a)\). The function \(y = 4\sin(x+\sin x)\) is a composition of continuous - functions (sine function and polynomial - like \(x+\sin x\)), so it is continuous everywhere.

Step2: Substitute \(x=\pi\)

We substitute \(x = \pi\) into the function \(y = 4\sin(x+\sin x)\). First, find the value of \(x+\sin x\) when \(x=\pi\). Since \(\sin\pi = 0\), then \(x+\sin x=\pi + 0=\pi\).

Step3: Calculate the sine value

Now, we find \(4\sin(x+\sin x)\) at \(x = \pi\). We have \(4\sin(\pi)\). Since \(\sin(\pi)=0\), then \(4\sin(\pi)=4\times0 = 0\).

Answer:

0