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Solve the given problems.$$\text { Find }\left.\lim _{\theta \rightarrow \pi / 2}(\sin \theta)^{\tan \theta} \quad \text { (See Exercise } 39 .\right)$$

Calculus 1 / AB

Chapter 27

Differentiation of Transcendental Functions

Section 7

L'Hospital's Rule

Derivatives

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we are asked to find the limit as X approaches. Prior to assign data raised the tangent data power. Now, if you plug in pi over to you right away, we find that we're left with the form one to the infinity power. And this is one of our indeterminate forms so we can apply low guitars rule once we rearrange this. So first step it's going to be letting why equal to the limit as X approaches pi over two of science data times tangent theater. Now we want to get rid of this tangent data as a power so we can take the natural log of both sides. So we can say that the natural log of why is equal to limit as X approaches pi over two of the natural log of science theater raised to the tangent fatal power. Now by our law algorithm rules, we can move this power the front so we can rewrite this as tangent theta times the natural log of science data. Now we want to rewrite this as an indeterminant product and to do that we can rewrite this as the natural log of sign theater divided by one over D. Tangent of theater. Now one over the tangent of data. This is the same as co tangent data. And now we have the form 0/0. So we can apply a little beetles rules the derivative of the numerator. We're left with co sign of theta divided by sign theater and our denominator. We're left with negative Kasey Kahne squared Ceta from here. We can rewrite this as co sign theater divided by sign theater divided by negative one over sine squared theta. Now what we can do here is we can pull this up and multiplied across. So we're left with co sign theater divided by science either divided by one over and square data. So that's just the same as times Science square theater. And we can't forget are negative. Now we're gonna cancel this science data in our denominator over there. That's gonna cancel power here. This is going to leave us with negative cosign theta sign fever. We're gonna copy this all the way down here. So we're left with negative cosign theta sign set up. And if we plug in pi over two we're left with zero. So we just solved for is the natural log of Y. So the natural love of why the secret is zero. And if you take the exponential of both sides we find that why is equal to E. 20 power Which is this one We know that. Why represents the limit Is experts five or two. So one is our final answer.

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