How do you find #lim costheta/(pi/2theta)# as #theta>pi/2# using l'Hospital's Rule?
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To find ( \lim_{\theta \to \frac{\pi}{2}} \frac{\cos(\theta)}{\frac{\pi}{2}  \theta} ) using L'Hôpital's Rule, we first take the derivative of the numerator and the derivative of the denominator separately, then evaluate the limit of their ratio.

Take the derivative of the numerator: [ \frac{d}{d\theta}(\cos(\theta)) = \sin(\theta) ]

Take the derivative of the denominator: [ \frac{d}{d\theta}\left(\frac{\pi}{2}  \theta\right) = 1 ]

Apply L'Hôpital's Rule: [ \lim_{\theta \to \frac{\pi}{2}} \frac{\sin(\theta)}{1} ]

Evaluate the limit: [ = \lim_{\theta \to \frac{\pi}{2}} \sin(\theta) ]

Since ( \sin(\theta) ) approaches 1 as ( \theta ) approaches ( \frac{\pi}{2} ), the limit is 1.
So, ( \lim_{\theta \to \frac{\pi}{2}} \frac{\cos(\theta)}{\frac{\pi}{2}  \theta} = 1 ).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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