Find the limit. Use l'Hospital's Rule where appropriate. If...

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{\theta \to \pi /2} \frac{1 - \sin \theta}{1 + \cos 2\theta} $

Key Concepts

Limits

Limits are used to describe the behavior of functions as the input approaches a particular value. They are a fundamental concept in calculus that formalizes the idea of a function's behavior near a specific point, even when the function might not be defined at that point.

Indeterminate Forms

Indeterminate forms arise when evaluating limits leads to expressions such as 0/0 or ?/?, which do not have a clear value without further analysis. Recognizing indeterminate forms is crucial because they signal that standard limit evaluation techniques may not directly apply, and methods like l'Hospital's Rule or algebraic simplifications might be necessary.

l'Hospital's Rule

l'Hospital's Rule is a method used to evaluate limits that result in indeterminate forms. It involves taking the derivatives of the numerator and denominator and then reassessing the limit. When the conditions for l'Hospital's Rule are met, it can simplify the process of finding the limit by turning an indeterminate form into one that is easier to evaluate.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variable where both sides are defined. These identities can be used to simplify expressions, especially when dealing with limits involving sine, cosine, and other trigonometric functions, allowing the limit to be evaluated more easily.

Transcript

0:00 Hello.

00:02 So here we consider the limit as theta approaches pi over two of one minus sine of theta divided by one plus cosine of two theta.

00:11 So we can use lobital's rule.

00:15 We have to first make sure we have an intertinent form that we can use lobital's rule on.

00:21 So we're going to go ahead and just do direct substitution first.

00:24 So if we do that, we have one minus sign of pi over two.

00:29 That's just going to be 1 minus 1.

00:31 Son of pi over 2 is 1.

00:32 So we have 1 minus 1.

00:34 And then divided by 1 plus cosine of 2 times pi over 2.

00:39 Well, 2 times pi over 2 is just pi.

00:41 So we have 1 plus cosine of pi.

00:44 Well, cosine of pi is negative 1.

00:45 So if we have 1 plus a negative 1, which gives us, well, 0 divided by 0.

00:51 So yes, we're in the form here, 0 over 0.

00:54 So low bettile's rule can be applied.

00:57 So we then go ahead and take.

00:59 The derivative of the top and divided by the derivative of the bottom.

01:03 So we then have the limit as theta approaches pi over two of, while the derivative of 1 minus sine of theta is going to be negative, negative cosine of theta...

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