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.

44. lim (x → 0⁺) (csc x - cot x + cos x)

Accepted Answer

Welcome back, everyone. Evaluated the following limit. Limit as X approaches 0 from the right of sequent X minus tangent X plus sine X. A says -1, B 0 C 1. Indeed, the limit does not exist. For this problem, let's go ahead and rewrite the limit. We get limit as X approaches 0 from the right of sequent X minus tangent X plus sine X. What we want to do is simply apply the properties of limits to get limit as X approaches 0 from the right of. 2 X minus limit as X approaches 0 from the right of tangent X. Plus limit as X approaches 0 from the right of sin X, right? So now our assumption is that each function is continuous and X equals 0, right? This is what we normally do as our next step, and we're going to perform direct substitution. So we're going to get sequence of 0, right? Now sequent is one divided by cosine, so we get 1 divided by. Cosine of 0. For the next part, we get tangent of zeros. We're subtracting tangent of 0, and we're adding sign of 0. Now, let's check if these are. Finite values. Remember, cosine of 0 is 1, so we get 1 divided by 1. Minus tangent of 0 is 0. It is continuous as well. And then we're adding sin of 0, which is 0. It is continuous. So now every function is continuous at 0, right, meaning the answer is 1 minus 0 + 0, which is 1. So the correct answer is option C. Thank you for watching.

Use l’Hôpital’s rule to find the limits in Exercises 7–52.44. lim (x → 0⁺) (csc x - cot x + cos x)

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Use l’Hôpital’s rule to find the limits in Exercises 7–52.
44. lim (x → 0⁺) (csc x - cot x + cos x)