Determine the following limits.

Question

\lim_{θ \to {\frac{π}{2}}^{+}} \frac{1}{3} \tan θ

Accepted Answer

Welcome back, everyone. In this problem, we want to find the limit as X approaches 3 pi from the right of 4/3 of the tangent of X divided by 2. A says it's infinity, B 0, C negative infinity, and D says it's 1. Now how are we going to find the limit of this expression? Well, first, let's make sure that we understand the function. The given function is 4/3 of the tangent of X divided by 2, and we are looking for the behavior of this function as X approaches 3 pi from the right. No recall, OK, that the tangent function, OK. Is the ratio of the sign function? To the cosine function. So in this case then, OK, that means we could rewrite or expression 4/3 of the tangent of X divided by 2, OK. As 4/3. Of the sine of x divided by 2 divided by the cosine of x divided by 2. No, that means then. That are limit, OK, as X approaches 3 from the 3 pi from the right is going to, if we replace our expression, OK, that means it's going to be this limit of 4/3 of the sign of X divided by 2. OK, divided by the cosine of X divided by 2 and the no, OK, we can put our constant outside of our limit. There's really then 4/3 of the limit as X approaches 3 pi from the right. Of the sine of x divided by 2 divided by the cosine of x divided by 2. Now what's the point here? Why did we rewrite our tangent function as the sign divided by the cosine function? Well, now we can analyze the behavior of the sine function and the cosine function to help us eventually figure out the limit as X approaches 3 pi from the right. Now, for the numerator, OK, for the sign of X divided by 2, OK, we know that the sign, OK, of 3 pi divided by 2 is equal to -1, OK. So now, what this tells us then is that as X approaches 3 pi from the right, OK, then that must mean the sign of X divided by 2 must be approaching -1. Now, since we found the limit of our numerator, let's try to think about the limit for our denominator, OK? So now for our denominator, the cosine of x divided by 2. If we think about it, the cosine of 3 pi divided by 2 is going to be equal to 0. I know for values slightly greater than 3 pi, X divided by 2 will be slightly greater than 3 pi divided by 2. And those are the values that will be approaching 3 pi from the right. Now this place is X divided by 2 in the 4th quadrant of the unit circle, and in the 4th quadrant, the cosine function is positive and approaches 0 as the value gets closer to 3 pi divided by 2. OK. So now what this tells us then is as X approaches 3 pi. From the right and the cosign of x divided by 2. Which is gonna be the cosine of 3 pi from the right divided by 2, is going to be approaching 0 from the right. In other words, it's going to be a small positive value. Know that we have the values that our assign and cosine functions are approaching for our limit, then no, this tells us then. That's our limit as X approaches 3 pi from the right. OK, we should multiply that by 4/3, my apologize apologies. So 4/3 of the limit as X approaches 3 pi from the right of the sign of X divided by 2, divided by the cosine of X divided by 2. Is going to be equal to 4/3 of -1 divided by 0, a small positive value, 0 being approached from the right. And now, in this case, because this represents a small positive value, if we divide 1 by a small positive value, the smaller the value, or sorry, the smaller the value gets, the closer our result is to negative infinity. That is, we're dividing -1 by that value. So that's gonna be 4/3 multiplied by negative infinity which equals negative infinity. Therefore, the limit. Is negative infinity. If we look back at our answer choices that there's a C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Determine the following limits.

$\lim_{θ \to {\frac{π}{2}}^{+}} \frac{1}{3} \tan θ$

Key Concepts

Limits

Tangent Function

One-Sided Limits

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