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_z→0 (tan 4z) / (tan 7z)

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. Determine the limit using Lahapital's rule if necessary. The limit as X approaches pi divided by 2 of cotangent of 5 X all divided by cotangent of 9 X. Awesome. So it appears for this particular problem, we're asked to evaluate this particular limit. So we're trying to determine what this value of this limit is. We're trying to determine the limit, and that is our final answer that we're ultimately trying to solve for. So with that in mind, let's read off our multiple choice answers to see what our final answer might be. A is 5 divided by 9, B is 9 divided by 5, C is 5, and D is 9. Awesome. So first off, in order to solve this particular problem, first off, let us use the direct substitution method to determine whether or not when we use the direct substitution method on our given limit, if it yields an indeterminate form, and if it yields an indeterminate form, then we can use Lahapatal's rule. So with that said, And let us recall once again really quickly here, what does indeterminate form mean? Well, an indeterminate form means if we perform direct substitution, and we get a value of 0 divided by 0, or if we get a value of infinity divided by infinity. So let's use direct substitution. So in this case, we're told that the limit as X approaches pi divided by 2. So in this case, X equals pi divided by 2. So we need to plug that value in for X directly, so we need to take cotangent of 5 multiplied by pi divided by 2. Fantastic, and then we need to divide it by cotangent of 9. Multiplied by pi divided by 2. So when we plug that into our calculator, we will find that it gives us an answer of infinity divided by infinity, which is indeed an indeterminate form. And because this direct substitution method yielded an indeterminate form, we can therefore know, as we should recall, we can apply the law hopital's rule now to evaluate this particular limit. So with that said, in order to solve this limit, because direct substitute, so since our direct substitution method did not work to give us an evaluate like to give us the answer for the evaluated limit, we have to use Lahoit's rule to evaluate this expression or for the expression for our limit because it's in an indeterminate form because of our direct substitution. So in order to get our final answer, we have to use Lahapa's rule. Awesome. So as we should recall and note, Laha Patal's rule states that for calculating limits with indeterminant forms, we need to differentiate the numerator and the denominator separately, where the numerator is the expression on the top of our division bar and the denominator is the bottom expression of our division bar. So we need to differentiate the numerator and denominator separately and repeat this process until we stop receiving an indeterminate form answer when we use the direct substitution method. So with that said, we need to, so therefore we need to apply Lajapata's rule once again to a solve for this limit. So in order to do that, once again, we need to find the derivatives of the numerator and denominator separately. So we need to differentiate the numerator and denominator with respect to X, which we could write this as D by DX because D by DX is a fancy way of saying taking the derivative with respect to X. So, the first derivative of the numerator, which as we should say is D by DX of cotangent of 5 X, and when we take the first derivative of the numerator, we will find that it is equal to Well, when we start to do it all, like when we start to evaluate, we will find that it's equal to negative cocant squad of 5 X multiplied by D by DX of 5 X, which is equal to -5 multiplied by cosicantsquad of 5 X. Fantastic. So now we need to differentiate the denominator with respect to X, so we need to take D by DX of cotangent of 9 X, which is equal to negative cos 2 of 9 X multiplied by D by DX of 9 X, which will give us an answer of negative 9 multiplied by cosicant squared. Of 9 X. Fantastic. So, with that said, moving right along here. We can now apply Lahapa's rule, so we, when we apply Lahapaal's rule, we'll be able to write that the limit as X approaches pi divided by 2. So when you take the limit as X approaches pi divided by 2 of cotangent of 5 x divided by cotangent of 9 X is equal to the limit, as X approaches pi divided by 2 of -5 multiplied by cosicant. Squared of 5 X all divided by -9 multiplied by cosicant squared of 9 X, fantastic. So Now, when we plug in a value of pi divided by 2, N for X, because that is our value for X that we need to plug in for the direct substitution method, you will get 5 multiplied by Cosicant squared of 5 divided by 2, divided by 9 multiplied by cosicant squared. Of 9, so 9 pi divided by 2. Fantastic. So when we plug that into our calculator, since we note that the negative symbols will cancel out and we will find that our final answer without a doubt will have to be equal to 5 divided by 9, and that is our final answer. Hooray, we did it. So the evaluated limit is going to be 59 or 5 divided by 9. And this corresponds with multiple choice answer letter A, which once again is 5 divided by 9. 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_z→0 (tan 4z) / (tan 7z)

Key Concepts

Limits

L'Hôpital's Rule

Trigonometric Functions

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