Determine the interval(s) for which the function is continuous. f ( x ) = sin x ( 2 + cos x ) f\left(x\right)=\frac{\sin x}{\left(2+\cos x\right)}

In this problem, we're asked to determine the intervals for which this function is continuous. Remember that when we're asked for these intervals, we need to give our answer in interval notation using our square brackets and our parentheses with those intervals contained inside. Now here, the function that we're given is f of x equals sine of x over 2 +cosine x. Now in order to find these intervals of continuity, we need to know if our function is discontinuous anywhere. Now how do we find where our function is discontinuous? Well, this is a rational function. Now even though there are trig functions here, this is still a rational function just involving trig functions. Now here since I have sine of x in my numerator, sine of x is continuous everywhere. I don't need to worry about anything there. But because this is a rational function, I do need to figure out if my denominator is ever equal to 0. So I'm gonna go ahead and take that denominator, 2 plus the cosine of x, and set it equal to 0. Now if I actually solve for x here, I can subtract 2 from both sides, giving me cosine x equals negative 2. And if I wanna solve for x here, I need to take the inverse cosine so x would be equal to the inverse cosine of negative 2. Now if you were to go ahead and type this into your calculator you would get an error because it's never going to happen that the cosine of x is equal to negative 2 because negative 2 is not within the domain of our cosine function. Remember that for our cosine function it goes between negative 1 and a positive one. So negative 2 is not contained within that interval. So there is nowhere that this denominator is going to be equal to 0 because this is not even possible. So because of that there are no discontinuities for this function. But because here we're asked to determine the intervals for which our function is continuous we need to state our answer using interval notation. Now because we don't have any discontinuities here that means our our function is continuous everywhere from negative infinity to positive infinity and that's our final answer here. This is where our function is continuous. Let me know if you have any questions, and I'll see you in the next one.

Determine the interval(s) for which the function is continuous. f ( x ) = sin ⁡ x ( 2 + cos ⁡ x ) f\left(x\right)=\frac{\sin x}{\left(2+\cos x\right)}

( − 1 , 1 ) (-1,1) ( − 1 , 1 )

( − ∞ , − 2 ) , ( − 2 , ∞ ) (-∞,-2),(-2,∞) ( − ∞ , − 2 ) , ( − 2 , ∞ )

1. Limits and Continuity

Continuity

Business Calculus

1. Limits and Continuity

Continuity

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Multiple Choice

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

$(-1,1)$

$(-∞,-2),(-2,∞)$

$(-∞,∞)$

The function is not continuous anywhere.

Verified step by step guidance

Step 1: Recall the definition of continuity. A function is continuous at a point if it is defined at that point, the limit exists at that point, and the value of the function equals the limit at that point.

Step 2: Analyze the given function f(x) = sin(x) / (2 + cos(x)). The numerator, sin(x), is continuous everywhere because it is a trigonometric function. The denominator, 2 + cos(x), is also continuous everywhere because cos(x) is a trigonometric function and adding 2 does not affect its continuity.

Step 3: Identify potential points of discontinuity. A rational function like f(x) is discontinuous where its denominator equals zero. Set 2 + cos(x) = 0 to find these points. Solve for x: cos(x) = -2. However, the range of cos(x) is [-1, 1], so cos(x) = -2 is not possible. Therefore, the denominator never equals zero.

Step 4: Conclude that the function f(x) is defined and continuous for all real numbers because there are no points where the denominator is zero, and both the numerator and denominator are continuous everywhere.

Step 5: State the interval of continuity. Since the function is continuous for all real numbers, the interval of continuity is (-∞, ∞).