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

Answer 1

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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Answer 2

#lim_(theta->0)(cos(7theta)-1)/sin(4theta)=0#

We first just try plugging in #0# to see what we get: #(cos(7*0)-1)/sin(4*0)=(1-1)/0=0/0#
This results in the indeterminate form #0/0#, which doesn't help us.
There is however a trick you can use to evaluate limits of this indeterminate form. It's called L'Hôpital's Rule. It basically says that if both the top and the bottom tend to #0#, we can take the derivative of both the top and bottom, knowing that the limit will be the same. Formally, this is: If #lim_(x->c) f(x)=0# and #lim_(x->c) g(x)=0# and #lim_(x->c) (f'(x))/(g'(x))=K# Then #lim_(x->c)f(x)/g(x)=K#
Applying this to our case, we get: #lim_(theta->0)(cos(7theta)-1)/sin(4theta)=lim_(theta->0)(d/(d\theta)(cos(7theta)-1))/(d/(d\theta)(sin(4theta))#
The derivatives can be computed using the chain rule: #d/(d\theta)(cos(7theta)-1)=-7sin(7theta)#
#d/(d\theta)(sin(4theta))=4cos(4theta)#
Now we put back into the limit and evaluate at #0#: #lim_(theta->0)(-7sin(7theta))/(4cos(4theta))=(-7sin(0))/(4cos(0))=(-7*0)/(4*1)=0/4=0#
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Answer from HIX Tutor

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.

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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