Determine the interval(s) for which the function is continuous.
$f(x)=\(\frac{sinx}{(2+cosx)}\)$

• A. $(-1,1)$
• B. $(-∞,-2),(-2,∞)$
• C. $(-∞,∞)$
• D. The function is not continuous anywhere.

Answer: (C)

Verified step by step guidance

1. First, identify the function given: f(x) = \(\frac{\sin x}{2 + \cos x}\). This is a rational function where the numerator is \(\sin\) x and the denominator is 2 + \(\cos\) x.
2. A rational function is continuous everywhere except where the denominator is zero. Therefore, we need to find where 2 + \(\cos\) x = 0.
3. Solve the equation 2 + \(\cos\) x = 0 to find the points where the function is not continuous. This simplifies to \(\cos\) x = -2.
4. Recall that the range of the cosine function is [-1, 1]. Since -2 is outside this range, there are no real values of x that satisfy \(\cos\) x = -2.
5. Since there are no values of x that make the denominator zero, the function f(x) = \(\frac{\sin x}{2 + \cos x}\) is continuous for all real numbers. Therefore, the interval of continuity is (-∞, ∞).