Trigonometry Limit Homework: Get Started Now!

[songoku]

Homework Statement

$$\lim_{x \to \frac{\pi}{3}} \frac{1-2 cos x}{\pi - 3x}$$

Homework Equations

trigonometry identity
properties of limit for trigonometry

The Attempt at a Solution

I have done several attempts but got me nowhere. I just need an idea to start.

Thanks

 

[mfb]

Do you know the rule of l'Hospital?

 

[songoku]

Do you know the rule of l'Hospital?

Yes and I am not allowed to use it.

 

[mfb]

Okay. Can you use a Taylor series?
Without any derivatives or approximations to the functions, it looks tricky.

 

[songoku]

Okay. Can you use a Taylor series?
Without any derivatives or approximations to the functions, it looks tricky.

I haven't learned it yet. I think I am only allowed to use trigonometry identities and limit properties

 

[Infrared]

Write $\lim_{x \to \frac{\pi}{3}} \frac{1-2 cos x}{\pi - 3x}$ as $\frac{2}{3}\lim_{x \to \frac{\pi}{3}} \frac{1/2- cos x}{\pi/3 - x}$. Notice that if you replace 1/2 with cos(π/3) you get something that looks like the definition of a derivative. It should be 2/3*cos'(π/3)

Edit: Do you know the derivative of cosine? If not, it is easy to calculate if you know the (1-cosx)/x and sinx/x limits.

 

[songoku]

Write $\lim_{x \to \frac{\pi}{3}} \frac{1-2 cos x}{\pi - 3x}$ as $\frac{2}{3}\lim_{x \to \frac{\pi}{3}} \frac{1/2- cos x}{\pi/3 - x}$. Notice that if you replace 1/2 with cos(π/3) you get something that looks like the definition of a derivative. It should be 2/3*cos'(π/3)

Edit: Do you know the derivative of cosine? If not, it is easy to calculate if you know the (1-cosx)/x and sinx/x limits.

I get it. Thanks a lot for your help