- #1
Alexx1
- 86
- 0
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?
I have to find the right limit from x --> 0
How can I solve this limit?
Last edited:
Mentallic said:Have you tried anything? How about simplifying the two logarithms and then using double angle formulae on the trigs.
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?
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?
Mentallic said:Yep!
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.
The limit of this expression as x approaches 0 is undefined or does not exist. This is because as x gets closer to 0, both sin(x/2) and cos(x/2) approach 0, causing the denominator to become 0 and making the expression undefined.
No, L'Hopital's rule cannot be applied in this case because the expression does not take on an indeterminate form (such as 0/0 or ∞/∞) as x approaches 0.
The limit of this expression as x approaches π/2 is -∞. This is because as x approaches π/2 from the left, sin(x/2) approaches 1 and cos(x/2) approaches 0, making the denominator approach 0 and causing the expression to approach -∞.
No, the limit will remain the same regardless of the base of the logarithms. This is because changing the base only results in a constant factor, which does not affect the overall limit.
This limit cannot be visualized graphically as it does not exist. As x approaches 0, the function becomes undefined, resulting in a vertical asymptote on the graph. However, the behavior of the function on either side of 0 can be observed by graphing the individual terms log(2sin(x/2)) and log(2(sin(x/2)+cos(x/2))).