Evaluate the Limit

Problem

(lim_x→π)(4)*sin(x+sin(x))

Solution

1. Identify the type of limit. Since the function 4*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, which states that (lim_x→a)(ƒ(x))=ƒ(a)

3. Substitute x=π into the expression.

(lim_x→π)(4)*sin(x+sin(x))=4*sin(π+sin(π))

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

4*sin(π+0)

5. Simplify the argument of the outer sine function.

4*sin(π)

6. Evaluate the final trigonometric value. Since sin(π)=0 the expression becomes 4*(0)

0

Final Answer

(lim_x→π)(4)*sin(x+sin(x))=0

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