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/2⁻ (π/2 - x) sec x

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to apply Lopital's rule to find the limit of the following function as X approaches pi divided by 2 from the left, and we're given the function F of X is equal to the quantity of 3 pi divided by 2 minus 3 X in quantity multiplied by of 3 X. We're given 4 possible choices as our answers. For choice A, we have 1, for choice B, we have 3. For choice C we have minus 1, and for choice D we have minus 3. Now we're asked to find the limit as X approaches pi divided by 2 from the left of our function F of X using Lopital's rule. So, our first step here is actually going to be to rewrite our function F of X. So we'll rewrite F of X as being equal to the quantity of 3 pi divided by 2 minus 3 X in quantity, divided by. Cosine of 3 X. So here we have just rewritten 2 of 3X as one divided by cosine of 3 X. So now we need to look at the limit, as X approaches pi divided by 2 from the left of F of X. And this is going to be equal to the limit, as X approaches pi divided by 2 from the left of the quantity of 3 pi divided by 2 minus 3X in quantity divided by the cosine of 3 X. So, at this point, we could try to evaluate this limit by direct substitution, and if we did, this would be equal to the quantity of 3 pi divided by 2, minus 3 multiplied by pi divided by 2 in quantity, divided by the cosine of 3 multiplied by pi divided by 2. And evaluate this expression, this would be equal to 0 divided by 0, which would be an indeterminate form. So what we're going to do is need to um apply Lopita's rule. In order to evaluate this limit, so we call that for Lopital's rule that we're going to take the derivative with respect to X of both our numerator and denominator, and then re-evaluate the limit. So our limit as X approaches pi divided by 2 from the left of F of X is going to be equal to the limit as X approaches pi divided by 2 from the left. Of the derivative with respect to X of the quantity of 3 pi divided by 2 minus 3 X. In quantity, divided by the derivative with respect to X. Of the quantity of cosine of 3 X. In quantity So, evaluating this derivatives, this would be equal to the limit, as X approaches pi divided by 2 from the left. Of the quantity, and in the numerator, I will have the derivative with respect to X of the quantity of 3 pi divided by 2 minus 3 X, and when I take that derivative, I get minus 3. The net is going to be divided by. The derivative with respect to X of cosine of 3 X. And so recall that when you take the derivative of the cosine function, you get back minus sign, but here we have to apply the chain rule and take the derivative of the argument as well. So we'll multiply our minus sign of 3 X by the derivative of 3 X, which is 3, so we in the denominator, we will have minus 3 multiplied by the sign of 3 X. So now we can actually simplify this expression. And we can divide both our numerator and denominator by -3, and so this is going to be equal to the limit as X approaches pi divided by 2 from the left of 1 divided by sin of 3 X. So now we can evaluate this limit by direct substitution. So this is going to be equal to 1 divided by the sign of 3 multiplied by pi divided by 2. Which is going to be equal to 1 divided by -1. Which is going to be equal to -1. And so that is going to be our answer for this problem. And so now what we need to do is compare our calculated answer to our answer choices, and we see that the correct answer choice corresponds to answer choice C. 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→π/2⁻ (π/2 - x) sec x

Key Concepts

Limits

L'Hôpital's Rule

Secant Function

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