Evaluate the Limit

Problem

(lim_x→π)(6)*sin(x+sin(x))

Solution

1. Identify the type of limit. Since the function 6*sin(x+sin(x)) is a composition of continuous functions (sine and addition), it is continuous everywhere.

2. Apply the direct substitution property for continuous functions by substituting x=π into the expression.

3. Evaluate the inner trigonometric function sin(π)

sin(π)=0

4. Substitute this value back into the expression to simplify the argument of the outer sine function.

6*sin(π+0)=6*sin(π)

5. Calculate the final value of the outer sine function.

6*(0)=0

Final Answer

(lim_x→π)(6)*sin(x+sin(x))=0

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