Evaluate the Limit

Problem

(lim_x→π)(9)*sin(x+sin(x))

Solution

1. Identify the limit type and the function. The function ƒ(x)=9*sin(x+sin(x)) is a composition of trigonometric functions, which are continuous everywhere on their domain.

2. Apply the direct substitution property for continuous functions. Since the function is continuous at x=π the limit is equal to the value of the function at that point.

(lim_x→π)(9)*sin(x+sin(x))=9*sin(π+sin(π))

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

9*sin(π+0)

4. Simplify the expression inside the outer sine function.

9*sin(π)

5. Calculate the final value. Since sin(π)=0 the expression becomes 9⋅0

0

Final Answer

(lim_x→π)(9)*sin(x+sin(x))=0

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