Evaluate the Limit

Problem

(lim_x→π)(sin(x+sin(x)))

Solution

1. Identify the type of function. The expression sin(x+sin(x)) is a composition of continuous functions.

2. Apply the property of limits for continuous functions, which allows the limit to be moved inside the function.

(lim_x→π)(sin(x+sin(x)))=sin((lim_x→π)(x+sin(x)))

3. Substitute the value x=π into the inner expression.

(lim_x→π)(x+sin(x))=π+sin(π)

4. Evaluate the trigonometric value sin(π)

sin(π)=0

5. Simplify the inner result.

π+0=π

6. Calculate the final sine value.

sin(π)=0

Final Answer

(lim_x→π)(sin(x+sin(x)))=0

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