Find the limits in Exercises 49–52. Write ∞ or −∞ where approp...

Question

Find the limits in Exercises 49–52. Write ∞ or −∞ where appropriate.

lim θ→0 (2 − cot θ)

Accepted Answer

Welcome back everyone. Find the limit. Limits theta approaches -5 divided by 2 of 3 plus tangent of theta. A says 3, B negative infinity, C infinity, and D does not exist. Now let's evaluate the limit. Limit as theta approaches -5 divided by 2 of 3 plus tangent of theta. As always, we, we begin with the direct substitution, assuming that our function is continuous at theta equals -5 divided by 2. So we get 3 plus we can rewrite tangent as the ratio between sine and cosine. So we get sine of negative pi divided by 2. Divided by cosine of negative pi divided by 2. This gives us 3 + -1 divided by 0, that's infinity, so 3 plus infinity leads to infinity. Now this is a difficult problem because essentially we cannot conclude that it is just infinity, right? Essentially it might be positive infinity, it might be negative infinity. So first of all, we have to conclude what type of sign it is, and then we also have to prove that this limit indeed exists because we're actually approaching a negative pi divided by T, not from the left, not from the right. We're approaching the point itself. So we have to apply the limit existence theorem. What we're going to do is simply check the limit as theta approaches negative, divided by 2 from the left of that function, 3 plus tangent of theta. And then we're going to check what happens if theta approaches-pi divided by 2 from the right of the same function, 3 plus tangent of theta. If they both lead to infinity with the same sign, then this is going to be our limit. If those signs are different, then the limit does not exist. So what we want to do is simply plot a coordinate plane that represents a unit circle. There is no necessity to draw the circle itself. We just want to recall that the y axis represents sin of X and the X-axis represents cosine of X. So, if they're both positive in the first quadrant, tangent is going to be positive, and the second quadrant cosine is negative and sine is positive, so the ratio is going to be negative. In the third quadrant, they're both negative, so tangent is going to Be positive and in the 4th quadrant, cosine as positive, sine is negative, so tangents is going to be negative. We are approaching a negative pi divided by 2, right? So if we start with the X axis, we're going. Clockwise, right? And essentially we can label negative pi divided by 2. As follows -5 divided by 2. Within our y axis. And now we are approaching -5 divided by 2 from the left and from the right. What does that mean? Well, in the first case, We can say that theta. is less than -5 divided by 2, because we're approaching it from the left, and in the second case, theta is greater than -5 divided by 2. Let's understand now that if we're approaching negative pi divided by 2 from the left, meaning data is less than -5 divided by 2, then we have a positive value of tangent, right? So in this case, our limit. Is going to be Positive. In affinity So let's write it down. Our first limit is positive infinity because we get 3 plus tangent of negative pi divided by 2 from the left, which gives us a positive tangent value, and we know that tangent as the approaches negative pi divided by 2 gives us infinity, right? So we have determined the sign. And now for the second case, if data is greater than negative pi divided by 2, we are in the 4th quadrant. And now we noticed that it gives us a negative value of tangent. So in this case, we're going to get negative infinity. What do we notice? Well, we get positive infinity for the limit from the left and negative infinity for the limit from the right. They are not equal to each other. They give us different signs for infinity, meaning the limit as state approaches 5 divided by 2 does not exist, right? So we're going to say that this limit does not exist based on the limit existence theorem since. The one-sided limits are not equal to each other and therefore the correct answer to this problem corresponds to the answer choice D. Thank you for watching.

Find the limits in Exercises 49–52. Write ∞ or −∞ where appropriate.

lim θ→0 (2 − cot θ)

Key Concepts

Limits

Cotangent Function

Indeterminate Forms

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