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.

14. lim (t → 0) sin 5t / 2t

Accepted Answer

Welcome back, everyone. Use Lapito's rule to find the limit as X approaches 0 of tangent of 3 X divided by X. For this problem, let's begin by rewriting the limit. We have our function, tangent of 3 X divided by X, and let's remember that the first step is to substitute X equals 0 for every X. We're going to get tangent of 3 multiplied by 0, which is tangent of 0 divided by 0, right? Tangent of 0 is equal to 0, so we get an indeterminate form 0 divided by 0, meaning we can apply Lapto's rule. Now Lato's rule says that we want to differentiate. The numerator, so we're going to take the derivative of tangent of 3 X and divide by the derivative of the denominator. Let's evaluate the derivative of the numerator. The derivative of tangent with respect to 3 X is squad of 3X. And because this is a composite function, we're going to apply the chain rule and multiply by the derivative of the inner function. So we're multiplying by the derivative of 3 X. And the derivative of X is 1, right? So now we can reduce. This form into limit as X approaches 0 of the derivative of 3 X is 3, we can factor out 3, and we're going to take the limit of squad of 3 X. Now, once again, let's perform direct substitution. We're going to get 3 multiplied by. Sequence squared can be rewritten as one divided by cosine squared, and we're taking 1 divided by cosine squared of. We multiplied by 0 gives us 0. Now, cosine of 0 is 11 divided by 1 squared is still 1, and 3 multiplied by 1 gives us a limit of 3. So that's our final answer. Thank you for watching.

Use l’Hôpital’s rule to find the limits in Exercises 7–52.14. lim (t → 0) sin 5t / 2t

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

14. lim (t → 0) sin 5t / 2t