# How do you find #\lim _ { \theta \rightarrow 0} \frac { \cos ( 7\theta ) - 1} { \sin ( 4\theta ) }#?

To find ( \lim_{\theta \to 0} \frac{\cos(7\theta) - 1}{\sin(4\theta)} ), use L'Hôpital's Rule, which states that if ( \lim_{x \to c} \frac{f(x)}{g(x)} ) is of the form ( \frac{0}{0} ) or ( \frac{\infty}{\infty} ), then ( \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} ), provided the latter limit exists.

Taking the derivatives of the numerator and denominator: [ \lim_{\theta \to 0} \frac{\frac{d}{d\theta}(\cos(7\theta) - 1)}{\frac{d}{d\theta}(\sin(4\theta))} ] [ = \lim_{\theta \to 0} \frac{-7\sin(7\theta)}{4\cos(4\theta)} ]

Substitute ( \theta = 0 ) into the expression: [ = \frac{-7\sin(0)}{4\cos(0)} ] [ = \frac{0}{4} ] [ = 0 ]

So, ( \lim_{\theta \to 0} \frac{\cos(7\theta) - 1}{\sin(4\theta)} = 0 ).

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When evaluating a one-sided 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 one-sided 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 one-sided 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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