Use l’Hôpital’s rule to find the limits in Exercises 7–52.
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Question

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

51. lim (θ → 0) (θ - sin θ cos θ) / (tan θ - θ)

Accepted Answer

Welcome back, everyone. Compute the following limit using Lapito's rule. Limit as X approaches 0 of sin x minus X cosine x divided by X minus tangent X. A says -1, B1, C0. Indeed, the limit does not exist. Let's begin this problem with direct substitution. We're going to rewrite the original limit, and we are going to substitute X of 0, right, into every X within our function. We begin with sign, so we got sine of 0. Minus X cosine x becomes 0 multiplied by cosine of 0. And we're dividing by 0 minus tension of 0. So now, sign of 0 is 0, we're subtracting 0 and dividing by 0 minus 0. Overall, this form is 0 divided by 0. And whenever we have this form, we can apply Lapito's rule immediately. So let's go ahead and do that. According to the rule, we want to find the derivative of the numerator. And the derivative of the denominator. Let's begin by differentiating. The numerator. We are going to get a limit as X approaches 0 off. The derivative of sine is cosine X. Then we're subtracting a product, so we want to apply the product rule. We're going to add a negative sign and in C, we're going to take the derivative of X multiplied by cosine X plus X multiplied by cosine x derivative. And now constraining the denominator, the derivative of X is 1, the derivative of tangent is C 2 X. Now, let's go ahead and simplify. By evaluating the remaining derivatives, so that we get limit as X approaches 0 of cosine X. Minus the derivative of axis once we subtracting cosine x. And we're going to subtract X multiplied by cosine x derivative. The derivative of cosine is negative signs, we can say that we're adding X X X. Notice that in the numerator, we can cancel out the cosine. And in the denominator, we have 1 minus 22 X. Once again, we're going to attempt direct substitution. So we get 0 multiplied by sign of 0, which is 0. And in the denominator, we're going to get 1 minus squad of 0. Now, sequence of 0 is 1, so again, we get 0. In the numerator and 0 in the denominator. Because we still have this form, we are going to perform differentiation once again, right? According to Lapito's rule. So now, let's find the derivative. We're going to get limit as X approaches 0. In the numerator, we have a product, so we're going to apply the product rule X X plus x x x divided by. In the denominator, the derivative of 1 is 0, so we can just factor out the negative sign and differentiates and 2 X using the chain rule. We're going to get to. Steak and eggs. Applying the power rule multiplied by the derivative of seek and which is, see and X tangent X. So overall this is to seek and squared tangent. To 2 squared X multiplied by tangent X. And now for the numerator, the derivative of X is one, the derivative of sine is cosine. Once again, we're going to attempt the direct substitution. This gives us negative. We have that negative sign in front, sign of 0 is 0, plus 0 multiplied by cosine of 0. It gives us 0 divided by 22 squared of 0, tangent of 0. This is 0 as well. So we still have 0 in the numerator and the denominator, right? So once again, we're going to apply. Lato's rule. Let's go ahead and do that. We're going to get negative limit as X approaches 0 of cosine x as the derivative of sine. Plus the derivative of X is one multiplied by cosine. Plus x multiplied by cosine x derivative, right? Which is negative X X X. cosine plus cosine gives us 2 cosine X. So we get 2 cosine x minus X X X. Divided by In the denominator, we have a product we can factor out too. And we're going to differentiate. So we're going to take the derivative of C and squared X. Which is to see. X, multiplied by sequent X derivative, we basically got 2 sequent squared X tangent X just as before. Plus 2 squared X multiplied by tangent derivative, right? The derivative of tangent is squad, so we're basically going to get sequence, the power of 4 of X. And then once again we're going to attempt direct substitution. We got negative, 2 cosine of 0, that's 2 multiplied by 1. Minus X sin x, where X is equal to 0, we got 0. Divided by 2 Which is multiplied by 2 2nd squad of 0. So, 2 of 0 is 1, we get 2 multiplied by 1, multiplied by tangent of 0. Well, tangent of 0 is 0, so we basically get 0 plus. And sequence to the power of 4 X, when X is 0 is 1, right? Well done, so now notice that we can cancel out. 2 and 2, right, because essentially we're subtracting 0 in the numerator and we get 1 divided by 1 which is 1 with the negative sign in front. Overall this is -1, which corresponds to the answer choice A, let's label it and thank you for watching.

Use l’Hôpital’s rule to find the limits in Exercises 7–52.51. lim (θ → 0) (θ - sin θ cos θ) / (tan θ - θ)

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Use l’Hôpital’s rule to find the limits in Exercises 7–52.
51. lim (θ → 0) (θ - sin θ cos θ) / (tan θ - θ)