Help with Limit: ((sin a)(sin 2a))/(1- cos a) as 'a approaches 0

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In summary, the limit of ((sin a)(sin 2a))/(1- cos a) as a approaches 0 is undefined. This can be solved using Taylor's Theorem and by rewriting sin 2a using the double angle identity. The answer to this problem is 4.
  • #1
charliemagne
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Can someone answer this problem for me? please!

What is the limit of ((sin a)(sin 2a))/(1- cos a) as 'a approaches 0.'

they say the answer is 4.

I really don't get it.

thank you

P.S.: Is sin a + cos a = 1
 
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  • #2


well we can't solve it for you but you have to think of limits like this (from what i remember)

sub in 0 and see what you get, and i think you will find its undefined at zero, so then you need to think what the curve is doing as it approaches zero. Put small numbers in and see where it is going :)
 
  • #3


One way to do it would use Tayor's Theorem.
 
  • #4


You want to somehow cancel out the 1-cosx in the denominator. sin a + cos a = 1 isn't the right identity but it's close, and that's one you'll need as well as the one with sin2a. Once you cancel 1-cosx, you can let a=0.
 
  • #5


Rewrite sin 2a using the double angle identity.
Multiply numerator and denominator by 1 + cos a. That will get you 1 - cos2 a in the denominator.

What "they" say about this limit is correct.
 
  • #6


Mark44 said:
Rewrite sin 2a using the double angle identity.
Multiply numerator and denominator by 1 + cos a. That will get you 1 - cos2 a in the denominator.

What "they" say about this limit is correct.

thank you

I got it

thank you for the 'hint'
 

FAQ: Help with Limit: ((sin a)(sin 2a))/(1- cos a) as 'a approaches 0

1. What is the limit of ((sin a)(sin 2a))/(1- cos a) as 'a approaches 0?

The limit of the given expression as 'a approaches 0 is 0.

2. How do you solve the limit of ((sin a)(sin 2a))/(1- cos a) as 'a approaches 0?

To solve this limit, we can use the trigonometric identity sin 2a = 2sin a cos a. Substituting this into the expression, we get (2sin^2 a cos a)/(1- cos a). Using the identity sin^2 a + cos^2 a = 1, we can simplify this to (2sin a cos a)/(1- cos a). As 'a approaches 0, sin a and cos a also approach 0, making the limit equal to 0.

3. Can we use L'Hopital's rule to solve the limit of ((sin a)(sin 2a))/(1- cos a) as 'a approaches 0?

Yes, we can use L'Hopital's rule to solve this limit. By taking the derivative of the numerator and denominator, we get (cos a)(2cos 2a)/(sin a) = 2cos 2a. As 'a approaches 0, cos 2a approaches 1, making the limit equal to 2.

4. What is the graph of the function ((sin a)(sin 2a))/(1- cos a) as 'a approaches 0?

The graph of the function as 'a approaches 0 is a straight line with a slope of 0 and a y-intercept of 0. This is because as 'a approaches 0, the function approaches 0, as we saw in the previous questions.

5. Why is it important to understand the limit of ((sin a)(sin 2a))/(1- cos a) as 'a approaches 0?

This limit is important because it is a fundamental concept in calculus and is used in various applications such as finding the slope of a curve, calculating instantaneous rates of change, and determining the behavior of functions at a certain point. It is also a building block for more complex limit problems and other calculus concepts.

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