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_x→0 csc x sin⁻¹ x

Accepted Answer

Below there today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Evaluate the limit. Use Lajapaal's rule if necessary. The limit as X approaches 0 of cosicant of X multiplied by the inverse tangent of X. Awesome. So it appears for this particular problem, we're ultimately trying to take this limit that is provided to us and we're trying to evaluate it. And that is our final answer that we're ultimately trying to solve for is what is the final answer of this evaluated limit. So with that said, let's read off our multiple choice answers to see what our final answer might be. A is -1, B is 0, C is 1. And D is DNE, which stands for does not exist. Awesome. So first off, in order to help us solve this problem, let us write cosicant of X in terms of sin of X. So therefore, we could take our limit, which is the limit as X approaches 0 of cosicant of X multiplied by the inverse tangent of X, and we could say that this is equal to the limit. And once again, we're rewriting Kosican of X in terms of sine. So that means we can rewrite our limit as the limit, as X approaches 0 of inverse tangent of X divided by sine of X. Fantastic. So now our first attempt we can use to solve this problem is we can use direct substitution, meaning we can just take what our limit is approaching, so X is approaching 0. So we just need a direct substitution in for 0, so we need to plug in the value for 0, N for X. So when we do that, we will find that the limit as X approaches 0 of inverse tangent of X divided by sin of X will give us. Inverse tangent of 0 divided by sin of 0, which will give us, so this once again, we're plugging in a value of 0 in for X. So when we plug that into our calculator, we will get a value of 0 divided by 0. And this gives us an indeterminate form. So this direct substitution method gives us an indeterminate form. And because it's an indeterminate form, that means we cannot use direct substitution. That means we need to recall and use La Hapital's rule in order to perform this evaluation for the specific limit. So using Lahapaal's rule, that means we need to take the limit, so we need to take our limit. When you take the limit, which once again is the limit as X approaches 0, of cosicant of X, which we're gonna write that in terms of signs, so we can take the limit as. X approaches 0 of inverse tangent of X divided by sine of X, and in order to do in order to use Lajapatal's rule, that means we need to take the first derivative of the numerator and the denominator, where the numerator is the top and the denominator is the bottom. So we need to take D by DX for the top and bottom. Of our division bar, so we need to take the first derivative of the numerator and the first derivative of the denominator, which our numerator is inverse tangent of X, and our denominator is sin of x. So when we take the first derivative of the numerator and the denominator, we will find that this is equal to the limit, as X approaches 0 of. And once again, when we take the first derivative of inverse tangent, we will get 1 divided by 1 plus X2, all divided by, and when we take the derivative of our denominator, we will get simply cosine of X, so cosine of X. Awesome. So now is the simple part. Now we can use our direct substitution method now, so we need to plug in a 0 in for our value of X. So we need to take the limit as X approaches 0 of 1 divided by 1 + 0 squad, all divided by cosine of 0. So when we plug that expression into our calculator, we will find when we use the direct substitution method after taking the first derivative of the numerator and denominator, we will find that our final answer is equal to one, and that's it. That is our final answer. Hooray, we did it. So our final answer without a doubt has to be multiple choice answer letter C, which is 1. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

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

lim_x→0 csc x sin⁻¹ x

Key Concepts

Limits

L'Hôpital's Rule

Cosecant and Inverse Sine Functions

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