Evaluate the Limit

Problem

(lim_x→π)(sin(x−sin(x)))

Solution

1. Identify the limit type and check for continuity. The function ƒ(x)=sin(x−sin(x)) is a composition of continuous functions (sine and subtraction).

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

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

3. Evaluate the inner trigonometric function. We know that sin(π)=0

sin(π−0)

4. Simplify the expression inside the outer sine function.

sin(π)

5. Calculate the final value of the sine function.

0

Final Answer

(lim_x→π)(sin(x−sin(x)))=0

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