Finding right limit of log(2sin(x/2))-log(2(sin(x/2)+cos(x/2))

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log(2sin(x/2))-log(2(sin(x/2)+cos(x/2))

I have to find the right limit from x --> 0

How can I solve this limit?
 
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Have you tried anything? How about simplifying the two logarithms and then using double angle formulae on the trigs.
 


Mentallic said:
Have you tried anything? How about simplifying the two logarithms and then using double angle formulae on the trigs.

If you simplify them you get:

log(sin(x/2)/(sin(x/2)+cos(x/2))

But if I fill in 0 than I get log(0) but that doesn't exist..
If I use a double angle formulae I also get log(0)
 


Well that's why it's a limit which suggests you need to figure out what the limit is as x approaches 0 from the right side. Where is it heading?
 


Mentallic said:
Well that's why it's a limit which suggests you need to figure out what the limit is as x approaches 0 from the right side. Where is it heading?

If I plot the function on my calculator, I thinks it's heading to -8, but I'm not sure
 


That's because your calculator isn't accurate enough or you can't see exactly what's happening really close (even closer than that) to 0.

You've already figured out that the answer is ln(0+) - the plus is there to show it's from the right side, not both sides of 0. 0- denotes from the left side.
What is the answer to ln(0+)? That's the same as asking what happens to the function y=ln(x) as x approaches 0 from the right?
 


Mentallic said:
That's because your calculator isn't accurate enough or you can't see exactly what's happening really close (even closer than that) to 0.

You've already figured out that the answer is ln(0+) - the plus is there to show it's from the right side, not both sides of 0. 0- denotes from the left side.
What is the answer to ln(0+)? That's the same as asking what happens to the function y=ln(x) as x approaches 0 from the right?

Is it -∞?
 


Yep! :smile:

To convince yourself, try every smaller positive values of x,

x=0.1
x=0.001
x=10-10

You'll see where it's headed.
 


Mentallic said:
Yep! :smile:

To convince yourself, try every smaller positive values of x,

x=0.1
x=0.001
x=10-10

You'll see where it's headed.

Thanks!
 
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