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→1⁻ (1-x) tan πx/2

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to apply Lopita's rule, define the limit of the following function as X approaches 2 from the left. If we're given the function G of X is equal to the quantity of 2 minus X in quantity multiplied by the tangent of the quantity of 3 pi X divided by 4 in quantity. We're given 4 possible choices as our answers. For choice A, we have 3 pi divided by 4. For choice B, we have minus 4 divided by the quantity of 3 pi. For choice C, we have 4 divided by the quantity of 3ie, and for choice D, we have minus 3 pi divided by 4. Now, we're asked to find the limit as X approaches to from the left of the function G of X using Lobital's rule. The first thing we're going to do is actually rewrite our function G of X. So we'll rewrite GFX as being equal to. The quantity of 2 minus X in quantity multiplied by stein of the quantity of 3 X divided by 4 in quantity, divided by cosine. Of the quantity of 3 pi X divided by 4 in quantity. So here we've just rewritten our tangent function in terms of sine and cosine. So now we need to find the limit. As X approaches to from the left of our function G of X. Which is going to be equal to the limit. As X approaches 2 from the left. Of the quantity of 2 minus X in quantity, multiplied by sine of the quantity of 3 pi X divided by 4 in quantity, divided by The cosine of the quantity of 3 pi X divided by 4 in quantity. Now if we try to evaluate this limit by direct substitution, this would be equal to quantity of 2 minus 2 multiplied by the sign of the quantity of 3 pi divided by 2 in quantity, divided by the cosine of 3 pi divided by 2 in quantity. And when we evaluate this expression, this would be equal to. In the numerator, 2 minus 2, so I'll have 0 in the numerator, divided by the cosine of 3 pi divided by 2 is 0, so we'll have 0 in the denominator. So we get the indeterminate form of 0 divided by 0. So to evaluate this limit, we're going to apply Lopital's rule, and so recall that Lopital's rule says that we'll need to take a derivative with respect to X of both our numerator and denominator, and then reevaluate our limit. So this is going to be equal to the limit. As X approaches 2 from the left. Of the derivative with respect to X of the quantity of 2 minus X in quantity, multiplied by the sine of the quantity of 3 pi X divided by 4 in quantity. That is divided by. The derivative with respect to X. Of cosine of the quantity of 3 pi X divided by 4 in quantity. So taking this derivative. This is going to be equal to. The limit As X approaches 2 from the left. And here in our numerator, when we take the derivative, we're going need to apply our product rule. So the first, um, if you recall from the product rule, we'll need to take the derivative of the first term of the product. So we'll take the derivative with respect to X of the quantity of 2 minus X. And when I take that derivative, I get -1, and that's going to be multiplied by the second term in our product, which is going to be sine of the quantity of 3 pi X divided by 4 in quantity. Then we'll have plus. The first terminal product, which is going to be the quantity of 2 minus X in quantity, and that gets multiplied by the derivative of the second terminal product, so we'll have the derivative with respect to X of the sign of the quantity of 3 pi X divided by 4. And so when we take the derivative of the sine function, we'll get a cosine function. So we'll have cosine. Of the quantity of 3 pi X divided by 4, but we'll need to apply our chain rule, so we'll need to multiply this by the derivative with respect to X of 3 pi X divided by 4, which is 3 pi divided by 4. And then that whole quantity is divided by the derivative with respect to X of cosine of the quantity of 3 pi X divided by 4 in quantity. And when we take the derivative of the cosine function, we get a minus sine function. So we'll have minus. Sign of the quantity of 3 pi X divided by 4. And again, we here have to apply our chain rule, so multiply this by the derivative with respect to X of 3 pi X divided by 4, which is 3 pi divided by 4. So now we can try to evaluate this limit by direct substitution. In doing so, this would be equal to minus 1. Multiplied by The sign Of the quantity Of 3 pie divided by 2 in quantity. Plus The quantity of 2 minus 2. In quantity, multiplied by 3 pi divided by 4, multiplied by The cosine of 3 pi divided by 2 in quantity, and that whole quantity is divided by Minus 3 pi divided by 4. Multiplied by the sign of the quantity of 3 pi divided by 2 in quantity. So this is going to be equal to. For the first time we'll have -1 multiplied by minus. One Since the sign of 3 pi divided by 2 is minus 1, then we'll have plus. 0, multiplied by. 3 pi divided by 4. Multiplied by 0, since the cosine of 3 divided by 2 is 0, and this whole quantity is divided by. The quantity of minus. 3 pie divided by 4. Multiplied by -1. So simplifying this expression, this is going to be equal to. 4 divided by the quantity of 3 high. So this here is the answer that we are looking for. And when we compare our calculated answer to our answer choices, we see that the correct answer choice corresponds to answer choice D. So I hope that this explanation has been useful, and I'll see you in the next video.

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

lim_x→1⁻ (1-x) tan πx/2

Key Concepts

Limits

l'Hôpital's Rule

Trigonometric Functions

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