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

Explanation:

Step1: Recall continuity property

If (y = f(x)) is continuous at (x = a), then (\lim_{x\rightarrow 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:

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