One-sided limit

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danishkid

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Feb 3, 2017
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Hi, geniuses!

I need your expertise once again now that I've encountered a problem that I can't really figure out and Google wasn't doing much to help.

I have to figure out what the limit of 1/x-cos(x)/sin(x) when x approaches pi-.

I know that the answer is infinity but when i try to calculate it i always get negative infinity.
My calculations are that 1/pi is a meaningless constant leaving us with -cos(x)/sin(x). cos(pi)=-1 and sin(pi)=0 so we have that - (-1) divided by an extremely small (negative, as it comes from -pi) number is equal to infinity. How is that possible? Are there any other ways to argue for the limit is infinity?

Thanks a lot! Any answer will be appreciated!
//DanishKid
 
You're oh so close.

sin(pi-) > 0
sin(pi+) < 0

Chew on that.
 
tkhunny said:
You're oh so close.

sin(pi-) > 0
sin(pi+) < 0

Chew on that.
Click to expand...

Maybe I'm misunderstanding this, but I look at the unit circle and see that if I go to pi+ (counterclockwise) then sin has a positive number as its slightly over the x-axis, but if I go to pi- I'll have to go clockwise around the circle and therefore end up in a negative number. Am I wrong on this?
 
danishkid said:
Hi, geniuses!

I need your expertise once again now that I've encountered a problem that I can't really figure out and Google wasn't doing much to help.

I have to figure out what the limit of 1/x-cos(x)/sin(x) when x approaches pi-.

I know that the answer is infinity but when i try to calculate it i always get negative infinity.
My calculations are that 1/pi is a meaningless constant leaving us with -cos(x)/sin(x). cos(pi)=-1 and sin(pi)=0 so we have that - (-1) divided by an extremely small (negative, as it comes from -pi) number is equal to infinity. How is that possible? Are there any other ways to argue for the limit is infinity?

Thanks a lot! Any answer will be appreciated!
//DanishKid
Click to expand...

Is your problem:

\(\displaystyle \displaystyle{\lim_{x \to {\pi}^{-}}\left [\frac{1}{x} - \frac{cos(x)}{sin(x)}\right ]}\)...................this is what you posted

or something else?
 
"pi-" has values less than pi. Approaching from an anti-clockwise direction if you must. Approaching from the left on a number line with typical orientation.
"pi+" has values greater than pi. Approaching from a clockwise direction if you must. Approaching from the right on a number line with typical orientation.

sin(pi-) > 0

If you look at a number line, rather than a circle, it will be more clear. The circle definition doesn't mean much for this problem, anyway. Leave it and move to the number line.
 
tkhunny said:
"pi-" has values less than pi. Approaching from an anti-clockwise direction if you must. Approaching from the left on a number line with typical orientation.
"pi+" has values greater than pi. Approaching from a clockwise direction if you must. Approaching from the right on a number line with typical orientation.

sin(pi-) > 0

If you look at a number line, rather than a circle, it will be more clear. The circle definition doesn't mean much for this problem, anyway. Leave it and move to the number line.
Click to expand...

I think I got it! Thank you very much! :)
 
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