Use l’Hôpital’s Rule to find the limits in Exercises 85–108.

Question

93. lim(x→0) (csc(x) - cot(x))

Accepted Answer

Welcome back, everyone. Find the limit using Lapto's rule. Limit as x approaches 0 of cos 2 X minus cotangent 2 X. A says 0, B1, C infinity, and D-1. For this problem, let's recognize that we have a difference, and because we want to apply Lapito's rule, we're going to rewrite the difference as a ratio. We have co-sequence to X minus cotangent to X. What we can do is simply use the definition of cosequent and cotangent, which gives us limit as X approaches 0. For cosequent, it is 1 divided by sin, so we get 1 divided by sin 2 X. And cotangent is the ratio between cosine and sine, we get cosine of 2 x divided by sin of 2 X. Notice that we already have a common denominator, so we can rewrite this limit as as x approaches 0 of 1 minus cosine of 2 X divided by sin 2 X. Now let's attempt the right substitution. We're going to get a 1 minus cosine. Of 2 multiplied by 0, which is cosine of 0, divided by sin of 2 multiplied by 0 gives us 0. In the numerator, we got 1 minus 1, which is 0. In the denominator, we also get 0 because sign of 0 is 0. Now that we have an indeterminate form 0 divided by 0, we can apply Lapito's rule, which sells. That we want to differentiate the numerator and the denominator. So we take 1 minus cosine of 2 x derivative and divide it by sin 2 x derivative. Let's find each derivative. We get limit as X approaches 0 of. The derivative of 1 is 0. It's a constant. We can factor out the negative sign. And the derivative of cosine is negative sign, right? We also want to apply the chain rules, so we get negative sign of 2 X. Multiplied by 2 x derivative, which is 2. We can also cancel out the two negative signs. And we can write 2 in front to get 2 signs of 2 X and the numerator. And the derivative in the denominator can be found using the chain rule again. So the the derivative of sinus cosine, we get cosine of 2 X multiplied by the derivative of 2 X, which is 2. Now notice that we can cancel out. The 2 And we are simply looking at. The ratio between sine and cosine. This is a limit as x approaches 0 of tangent of 2 X. And now let's attempt direct substitution once again, we're going to get tangent of 2 multiplied by 0, which is 0. Intentions of 0 is equal to 0, meaning the right answer for this problem corresponds to the answer choice A. Thank you for watching.

Use l’Hôpital’s Rule to find the limits in Exercises 85–108.93. lim(x→0) (csc(x) - cot(x))

Watch next

Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
93. lim(x→0) (csc(x) - cot(x))