109-112 {Use of Tech} Calculating limits The following limits ...

Question

109-112 {Use of Tech} Calculating limits The following limits are the derivatives of a composite function g at a point a.

b. Use the Chain Rule to find each limit. Verify your answer by using a calculator.

$\lim_{x \to 0} \frac{\sqrt{4 + \sin (x)} - 2}{x}$

Accepted Answer

Welcome back, everyone. In this problem, we want to apply Lopita's rule to find the limit of the following function as X approaches 0, where F of X equals the square root of 9 plus sine X minus 3 all divided by 2 X. A says it's a quarter, B 1/3, C 1/12, and D says it's 19th. Now, before we can apply Lapita's rule, OK, we first have to establish that direct substitution would not work. In other words, that it would lead to an indeterminate form. So let's try to do that. Let's think about what would happen. So let's consider what would happen if we find the limit as X approaches 0 of F of X. No, that means we'll be finding the limit as X approaches 0 of the square root of 9 plus sine X. -3 all divided by 2 X. Which is gonna be the square root of 9 + 0 minus 3 divided by 2 multiplied by 0. And knowing our numerator, 0 equals 0, the square root of 9 is 3, and 3 minus 3 would be equal to 0. In our denominator 2 multiplied by 0 equals 0. I know we've run into a problem because we cannot divide by 0. Therefore, the direct substitution of X equals 0 would not work. Thus, that leads us to the potential of having to use Lapita's rule. Now, do you remember what Lapita rule says? We'll recall that it basically tells us that if we are finding the limit as x approaches some value a of a quotientu divided by V where U and V are functions of X, then that's going to be the same thing as finding the limit as x approaches A of the derivative of u with respect to X divided by the derivative of V with respect to X. So, in other words, we can differentiate our numerator and denominator and then find the limit. So let's see if we can do that, OK? So let's first let U equal to the square root of 9 plus sine X. -3 OK. And then we're going to let V be equal to 2 X. Now, let's start with you. Let's differentiate you. Now, notice here for you, OK. Uh, by differentiating it, here we have a composite function, the square root of 9 plus sine X that we differentiate, and we also need to differentiate minus 3. Now, -3 is a constant, so we know its derivative is going to be 0. So DUDX is going to be the derivative of our composite function. The square root of 9 plus sine X K. Plus, the derivative of -3, which is 0. OK. Now, what is the derivative of our function here? Well, we can apply the chain rule. And the chain rule basically tells us that if we are going to differentiate uh our our term here then it would probably first be easier to write or root as a power so that's 9 + X raise to the power of a half and not to differentiate it with the chain rule, then we are going to first bring down our power, which is a half. Multiply it by our original function minus a power of 1. So that's going to be multiplied by 9 + X, OK, or auto function raised to 1/2 minus 1, which is negative 5. And then we're going to multiply that by the derivative of sine X, which is going to be the cosine of x. Now if we simplify this expression, OK. First, we can write 9 + sine x in our denominator. So that's gonna be 1 divided by 2 multiplied by 9 + sine x-rays to the power of a half or the root of 9 + sine X. And all of this is going to be multiplied by the cosine of X. So actually, we could just write the cosine of X in our numerator here, OK? So this is the derivative of our numerator. No, on the other hand, OK, let me just separate this here. On the other hand, if we let V be equal to 2 X, then the derivative of V with respect to X is going to be equal to 2, based on the power rule. Know that we are differentiated while of it's rule, that must mean then that the limit as x approaches 0 of F of X is going to be equal to the limit as X approaches 0 of. Or a term here which is gonna be one. Or numerator, let me write that. The cosine of X divided by 2 multiplied by the square root of 9 plus sine X. OK. All divided by 2 or multiplied by a half, which we can write as the cosine, OK. The limit as X approaches 0, of the cosine of X divided by 4 row 9 plus sine X. Now we can go ahead and solve. So no, that's gonna be equal to the cosine of 0 divided by 4 multiplied by the root of 9 plus 0. The cosine of 0 is 1, the sign of 0 is 0, so that's equal to 4 multiplied by the square root of 9, and the square root of 9 is 34 multiplied by 3 equals 12. Thus, the limit of FF X as X approaches 0 by using Lapita's rule would give us 1/12. If we look back at our answer choices, that means C is the correct answer. Thanks for watching everyone. I hope this video helped.

109-112 {Use of Tech} Calculating limits The following limits are the derivatives of a composite function g at a point a.
b. Use the Chain Rule to find each limit. Verify your answer by using a calculator.
$\lim_{x \to 0} \frac{\sqrt{4 + \sin (x)} - 2}{x}$

Key Concepts

Limits

Chain Rule

Derivatives

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