How do you find #lim sin(2theta)/sin(5theta)# as #theta>0# using l'Hospital's Rule?
According to L'Hospitol's Rule
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To find ( \lim_{\theta \to 0} \frac{\sin(2\theta)}{\sin(5\theta)} ) using l'Hospital's Rule:

Apply l'Hospital's Rule:
 Differentiate both the numerator and the denominator with respect to ( \theta ).
 Compute the limit of the resulting expression as ( \theta ) approaches 0.

Differentiate the numerator:
 ( \frac{d}{d\theta} \sin(2\theta) = 2\cos(2\theta) )

Differentiate the denominator:
 ( \frac{d}{d\theta} \sin(5\theta) = 5\cos(5\theta) )

Substitute into the limit expression:
 ( \lim_{\theta \to 0} \frac{2\cos(2\theta)}{5\cos(5\theta)} )

Evaluate the limit:
 Substitute ( \theta = 0 ) into the expression: ( \frac{2\cos(0)}{5\cos(0)} = \frac{2}{5} )

The limit of ( \frac{\sin(2\theta)}{\sin(5\theta)} ) as ( \theta ) approaches 0 is ( \frac{2}{5} ).
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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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