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One-sided limit

danishkid

New member
Joined
Feb 3, 2017
Messages
10
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
 

tkhunny

Moderator
Staff member
Joined
Apr 12, 2005
Messages
9,701
You're oh so close.

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

Chew on that.
 

danishkid

New member
Joined
Feb 3, 2017
Messages
10
You're oh so close.

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

Chew on that.
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?
 

Subhotosh Khan

Super Moderator
Staff member
Joined
Jun 18, 2007
Messages
18,089
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
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?
 

danishkid

New member
Joined
Feb 3, 2017
Messages
10
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?
Yes! That's my problem
 

tkhunny

Moderator
Staff member
Joined
Apr 12, 2005
Messages
9,701
"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.
 

danishkid

New member
Joined
Feb 3, 2017
Messages
10
"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.
I think I got it! Thank you very much! :)
 
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