60–81. Limits Evaluate the following limits. Use l’Hôpital’s R...

Question

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.

lim_Θ→0 2Θ cot 3Θ

Accepted Answer

Welcome back, everyone. Evaluate the limit. Use laps rule if necessary. Limit SX approaches 0 of 9 X cotangent of 17 X. We're given 4 answer choices AS 1 divided by 17, B9, C 9 divided by 17, and D 17 divided by 9. So let's rewrite the given limit. Where given limit as X approaches 0. Of 9 X cotangent of 17 X. Now what we have to do is simply apply the right substitution. 9 multiplied by 0 is 0, and we're multiplying that by cotangent of 0, right? So now for cotangent of 0, it can either be positive or negative infinity depending on which side X approaches 0 from, so we can get positive or negative infinity. In any case, so far we are getting 0 multiplied by infinity, right? So what we're going to do is simply consider rewriting this limit in the form of 0 divided by 0 or infinity divided by infinity. So we're going to get the limit as X approaches 0. We can factor out 9. And in the numerator we have X. Now for the denominator, why we're even getting it, well, we have to remember that cotangent of X is just the reciprocal of tangent, right? So cotangent of X is one divided by tangent of X. So in this case we're getting X divided by tangent of 17 X. And in this case, this is where we are going to get an indeterminate form of 0 divided by 0. Now, why is this the case? Well, this limit exists because from the left, if X approaches 0 from the left, then X is going to approach 0 from the left, right? If it's from the right, X is going to approach 0 from the right for tangent, that's equivalent, right? If X approaches 0 from the left, tangent also approaches 0 from the left. If X approaches 0 from the right, tangent approaches 0 from the right as well. So we get 0 divided by 0 in any case, so the limit exists, right? And since this is an indeterminate form, we can simply apply Lapito's rule, which gives us 9. Limit as X approaches 0 and Lapito's rule says that we can differentiate the numerator and the denominator if we get that in the terminate form. So in the numerator, we get The derivative of X and in the denominator we're taking the derivative of tangent of 17 X. Now let's evaluate each derivative. We get 9 limit x approaches 0 of 1, because the derivative of X is 1, the derivative of tangent is C2, so we're getting squad of 17 X. But we have an inner function 17X, so we also have to multiply that by the derivative of 17 X. And the derivative of 17 X is 17, right? So now we can combine our constants. We got 9 divided by 17 because we have 17 in the denominator. 9 divided by 17, limit as X approaches 0 of 1 divided by 2 squad of 17 X. Now let's go ahead and apply the direct substitution once again. We're getting 9 divided by 17, multiplied by 1 divided by squad of 0. Now, squad or 2 of 0 is 1, so squad is also 11 divided by 1 gives us 1. And this means that the final answer is 9 divided by 17. It is a finite number, meaning we have successfully evaluated the final answer, which corresponds to the answer choice C. 9 divided by 17. Thank you for watching.

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.

lim_Θ→0 2Θ cot 3Θ

Key Concepts

Limits

L'Hôpital's Rule

Trigonometric Functions

Watch next