Evaluate the following limits. Use l’Hôpital’s Rule when it is...

Question

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

lim_x→π/2⁻ (π - 2x) tan x

Accepted Answer

Welcome back, everyone. In this problem, we want to evaluate the limit as X approaches pi from the right of pi minus X multiplied by the cotangent of X using Lapita's rule if necessary. A says it's -1, B negative route 2 divided by 2. C Route 3 divided by 2, and D 2. Now to evaluate our limit, let's not jump to Lapita's rule yet. Let's first test if we can figure out the limit without it resulting in an indeterminate form. No, that means we're going to substitute X equals pi from the right to see what form the expression takes. So by direct substitution, OK. By direct substitution. If we substitute X approaching pi from the right into the expression pi minus X, then as X approaches pi from the right. Pi minus X will approach 0 from the left. OK? Likewise, As X approaches, well, for a term, the cotangent here, since the cotangent of X. Is equal to or is the ratio of the cosine of X to the sine of X. Then as X approaches pie from the right, we know that sin X right beside it here. Sine X, approaches 0 from the left and cause X, X approaches -1. So in that case, this means that the cotangent of X approaches infinity. So in that case then. By direct substitution. Then i minus X multiplied by the cotangent of X. I shouldn't write that in red. Let me keep that in black here. This means that our limit by direct substitution. Is going to be equal to 0 multiplied by infinity and this will result in an indeterminate form. Indeterminate. So now that we've established that it's indeterminate, indeterminate, then we can apply Lapita's rule. What does Lapita's rule tell us? Well, Here, OK, then. What we know is that. Recall, OK, we can say recall here that if the limit, OK. The limit as X approaches some value of FX divided by GX are caotient. Is indeterminate, OK. Then we can rewrite that limit, take the limit as X approach is A of F of X divided by G of X. And rewrite it as the limit or it's going to be equal to the limit as x approaches a of the derivative of F of X divided by the derivative of G of X. Now initially our problem is not written as a quotient, but recall like we saw earlier, we could rewrite pi minus x multiplied by the cotangent of X. As pi minus X multiplied by the cosine of X divided by the sine of X, and it would still make sense for us to use Lapita's rule because our numerator and denominator here would eventually value it to zero if we're finding the limit as X approaches pi from the right. OK. So now, if we can differentiate our numerator and denominator here with respect to X, then we can apply Lapita's rule. So let's let FFX. Be equal to minus X, multiplied by the cosine of X, OK. And let's let Geoff fix. Be equal to the sign of X. Now for FF X we can differentiate this using the product rule of differentiation, and we're going to get the negative cosine of X plus pi minus X multiplied by negative sine X. And if we evaluate that a bit further, it gives us negative plus X minus pi minus X multiplied by sin X. OK? This is the derivative of the numerator. On the other hand, the derivative of the denominator sine x is going to be cos X. Know that we have that by applying Lapita's rule, then we can we or limit, sorry, is going to be equal to or sorry, the limit as X approaches pi from the right, OK, minus. Cost x minus pi minus X. X. Divided by the cosine of X. Now what is our limit going to be here? Well, no, we can tell that as X approaches pi from the right, then i minus X multiplied by sin x. Approaches 0. And the cosine of X approaches -1, and these are all the terms in our numerator and denominator here. So what that means is that our limit is going to be equal to the limit as X approaches pi from the right of negative, -1 minus 0 divided by -1. That's 1 divided by -1, which equals -1. Thus, by Lapita's rule, the limit as X approaches pi from the right of pi minus X cotangent X is going to be equal to -1. A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→π/2⁻ (π - 2x) tan x

Key Concepts

Limits

l'Hôpital's Rule

Trigonometric Functions

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