Is there a value of c that will make

f(x) = {...

Question

Is there a value of c that will make

f(x) = { (sin²(3x)) / x², x ≠ 0

c, x = 0

continuous at x = 0? Give reasons for your answer.

Accepted Answer

Welcome back everyone. Find the value of a that ensures the function HX is continuous at X equals 0. A says 2, B4, C8, and D 16. According to the context of this problem, H of X is a piecewise function. It is defined as sin to the power of 4 of 2 x divided by X to the power of 4 if x is not equal to 0 and it is equal to a if x equals to 0. So now let's recall that if a function is continuous, then the value of the function at a point in this case our point of interest is X equals 0 because It changes the behavior of the function from sign to a constant function, right? So we want to specifically focus on X equals 0, and we have to recall that the continuity condition tells us that the value of H at that point of 0. Must be equal to the limit as x approaches 0 of H of X. So we're going to use this condition. First of all, let's valuate H at 0.0 according to the context of the problem. If X is equals 0, then the value of the function is A, right? And now our limit. As X approaches 0 of H of X is going to be. The limit as X approaches 0 of sin to the power of 4 of 2 X divided by X to the power of 4, and we want to set it equal to A. So let's solve for the limit. What we're going to do is first we'll use. The power law or limits, which gives us a limit as x approaches 0 of sin of 2 x divided by X, all of that raises the power of 4. We can notice that our numerator and the denominator has an exponent of 4, so we can basically take it out, right? And from here we have to report that the limit. As X approaches 0 of sin of x, divided by X is equal to 1, and we're going to use this theorem. So what we can do is simply set limit as x approaches 0. Of sin of 2 Xs divided by X. To limit as x approaches 0 of sin of 2 x divided by 2 X, right, we essentially want to have the same angle as our denominator, but because we're adding that 2 in the denominator, we also have to add one half in front. 1/2 multiplied by 2 gives us 1, so we are not changing the original function. And now what we can do is simply notice that division by 1/2 gives us 2 in the numerator, right? So we're going to get. Limit as X approaches 0 of 2 of 2 X. Divided by 2 Xs. All of that is raise the power of 4. Now using the properties of limits, we can seek out the constant. Which gives us 2. Limit as X approaches 0 of sign of 2 Xs divided by 2 Xs. All of that raise the power of 4. And now we can raise 2 to the power of 4, which is going to give us 16. Multiplied by limit as X approaches 0 of sign of 2 X divided by 2 Xs. Raise the power of 4. Now, if X approaches 0, then 2 X is also going to approach 0. So now we are consistent with the theorem, right? And we know that our limit is equal to 1. Meaning we get 16 multiplied by 1 which is 16, and going back to our previous condition, this must be the value of a to ensure that H of X is continuous at X equals 0, meaning the correct answer to this problem corresponds to the answer choice B. A is equal to 16. Thank you for watching.

Is there a value of c that will make

f(x) = { (sin²(3x)) / x², x ≠ 0
c, x = 0

continuous at x = 0? Give reasons for your answer.

Key Concepts

Continuity at a Point

Limit of a Function

L'Hôpital's Rule

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