17–83. Limits Evaluate the following limits. Use l’Hôpital’s R...

Question

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→π (cos x +1 ) / (x - π )²

Accepted Answer

Below there. Today we're gonna 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. Calculate the value of the given limit. Apply Lahapeal's rule if necessary. The limit as x approaches pi divided by 2 of pi divided by 2 minus x to the power of 2 all divided by 1 plus cosine of 2 X. fantastic. So it appears for this particular problem, we're asked to evaluate this particular limit. So we're ultimately trying to calculate the value of this limit, and that is our final answer that we're trying to solve for. So now that we know that we're trying to evaluate this limit, let's read off our multiple choice answers to see what our final answer might be. A is 0, B is 1/4, C is 1/2, and D is 4. We could also say that B is 1 divided by 4 and C is 1 divided by 2. Awesome. So with that said, we're given a hint by the prom itself it says apply Lajapatal's rule if necessary, and since we're given this hint, we should be able to recall a note that we're allowed to use Lajapatal's rule if when we use the direct substitution method, we will receive an indeterminate form as an answer. And if we receive an indeterminate form as an answer after direct substitution. Then we're allowed to use Lahapatal's rule. So what's an indeterminate form? Well, one of our indeterminate forms, as we should recall, is 0 divided by 0 or infinity divided by infinity. Fantastic. So, let's try the direct substitution method to see if our limit right now when we use direct substitution into our given limit if we will receive an indeterminate form as a final answer. So in this case, we're told that X approaches pi divided by 2, so we could say that X is equal to pi divided by 2. So if we plug in a value of pi divided by 2 in for X, we will get that pi divided by 2. Minus pi divided by 2. All squared, divided by, and we need to take this divided by. 1 plus cosine of 2 multiplied by pi divided by 2. Fantastic. So when we plug that into our calculator, we will find that it will be equal to 0 divided by 0, which is an indeterminate form. So therefore, as a result, since when we use the direct substitution method and it yielded an indeterminate form as an answer, we have to recall and use Lahoital's rule in order to evaluate this particular limit. So, with that said, as we should recall a note, Lahapital's rule states that for calculating limits with indeterminate forms, we need to differentiate the numerator and denominator separately, and we need to repeat this process until we stop receiving an indeterminate form as an answer when we use the direct substitution method. So with that said, let's quickly recall that the numerator is the expression on top of our division bar, whereas the denominator is the expression on the bottom of our division bar. So, with that said, we need to apply La Hait's rule in order to solve this particular limit. So in order to do that, once again, we need to take the derivatives of the numerator and denominator with respect to X. So let's start with the numerator. So let's take D by DX of pi divided by 2, minus X alter the power of 2. Fantastic. And as you can see, for this particular derivative that we need to take, we need to recall and use the chain rule in order to differentiate the numerator. So with that said, so let me do that and noting once again that D by DX is just a fancy way of saying we're taking the derivative with respect to X. So when we start to perform the differentiation, we will find that this is equal to 2 multiplied by pi divided by 2 minus X, multiplied by D by DX of pi divided by 2 minus X, and this will be all equal to. 2 multiplied by pi divided by 2 minus X. So it's pi divided by 2 minus X multiplied by -1, which will be all equal to -2 multiplied by pi divided by 2 minus X. Fantastic. So now we need to differentiate the denominator with respect to X. So we need to take D by DX of 1 plus. So D by DX of 1 plus cosine of 2 X. Fantastic. And this will be equal to 0 minus sign of 2 X multiplied by D by DX of 2 X, which is going to be all equal to -2 multiplied by sin of 2 X. Fantastic. So, moving right along here, we now need to evaluate the limit at this particular stage, so we need to take the limit as X approaches, pi divided by 2, so we need to take the limit as X approaches pi divided by 2. Of I divided by 2, so i divided by 2 minus X2. All divided by one plus. Cosine of 2 X, fantastic, is equal to the limit as X approaches pi divided by 2. Of Negative. 2 multiplied by divided by 2 minus X, so pi divided by 2 minus X all divided by negative, 2 multiplied by sin. Of 2 X, so negative 2 multiplied by sin of 2 X. Fantastic. So, We can say that this is all equal to the limit, as X approaches pi divided by 2 of, and we're running out of rooms, so let's move this down below, shall we? So we have a little bit more room to work with. So, we could say that the limit as X approaches pi divided by 2 of, pi divided by 2, minus X all divided by. Sign of 2 x, and once again we can receive this answer by doing a little bit of simplifying, and this will be all equal to when we plug in our value for X, which in this case X is going to be equal to pi divided by 2, so we're using direct substitution now. So this will be equal to pi divided by 2 minus pi divided by 2, all divided by sin of 2 multiplied by pi divided. So it's 2 multiplied by pi divided by 2. And when we plug that into our calculator, unfortunately, we get an indeterminant form once again, which is 0 divided by 0. So that means we have to do, we have to do the differentiation of the numerator and denominator again. Which is unfortunate, but we got this. It's no harm, no foul. So now we need to apply Lahapatal's rule again, and we need to take. Our limit, which right now we need to take the limit as X approaches pi divided by 2 of pi divided by 2 minus X, divided by sin of 2 X, and we need to differentiate the numerator and denominator. So, when we differentiate the numerator, we will take D by DX of pi divided by 2 minus X, which is going to be equal to 0 minus 1, which gives us -1, and doing that to the denominator now, we need to differentiate the denominator with respect to X, so D by DX of Sign of 2 X, which is going to be equal to cosine of 2 X multiplied by D by DX of 2 X, which is going to be equal to 2 multiplied by cosine of 2 X. Fantastic. So now we need to evaluate the limit again. So when we do that, we will find that the limit as X approaches pi divided by 2 of pi divided by 2, so you take. I divided by 2 minus X divided by sin of 2 X is equal to the limit as X approaches pi divided by 2 of. -1 divided by 2 multiplied by cosine of 2 X, which is going to be equal to now. So now we need to plug in our value for the value for X since we're using direct substitution at this stage, so we need to take -1. Or you could pull out the negative to the front, so let's do that so we can. I have a more clear answer, so negative, 1 divided by 2 multiplied by cosine of 2 multiplied by pi divided by 2. Fantastic. So, when you plug this into your calculator, you will find that we will receive an answer of, or if you want to do it by hand a little bit to stretch it out, you'll get -1 divided by 2 multiplied by -1. Because this cosine of 2 multiplied by pi divided by 2 will give you -1, so it's gonna be 2 multiplied by -1, which will be when you simplify and write your final answer infraction form, you'll get a final answer of 1/2, and that's it. We've officially solved for this problem. Hooray, we did it. So, because this is not 0 divided by 0 or infinity divided by infinity, we've officially evaluated the limit. So looking at the answers, are all of our multiple choice answers, it appears our final answer has to be multiple choice answer letter C, which once again is one half. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→π (cos x +1 ) / (x - π )²

Key Concepts

Limits

L'Hôpital's Rule

Continuity

Watch next