Finding Limits


In Exercises 9–24, find the limit or explain why it does not exist.




lim x →π sin (x/2 + sin x)

Verified step by step guidance

1. Identify the type of limit problem: We need to find \( \lim_{{x \to \pi}} \sin\left(\frac{x}{2} + \sin x\right) \). This involves a trigonometric function, so we should consider the behavior of the function as \( x \) approaches \( \pi \).
2. Substitute \( x = \pi \) into the expression inside the sine function: Calculate \( \frac{\pi}{2} + \sin(\pi) \). Since \( \sin(\pi) = 0 \), this simplifies to \( \frac{\pi}{2} \).
3. Evaluate the sine of the resulting expression: We need to find \( \sin\left(\frac{\pi}{2}\right) \). Recall that \( \sin\left(\frac{\pi}{2}\right) = 1 \).
4. Consider the continuity of the sine function: Since the sine function is continuous everywhere, the limit of \( \sin\left(\frac{x}{2} + \sin x\right) \) as \( x \to \pi \) is simply the sine of the limit of the expression inside, which we found to be \( \frac{\pi}{2} \).
5. Conclude the limit: Therefore, the limit \( \lim_{{x \to \pi}} \sin\left(\frac{x}{2} + \sin x\right) \) is equal to \( \sin\left(\frac{\pi}{2}\right) = 1 \).