Why is the limit of cot(x) approaching pi from the negative side -infinity?

[shocklightnin]

Homework Statement

lim x->pi- cot(x)

Homework Equations

cot(x) = cos(x)/sin(x)

The Attempt at a Solution

so substituting pi into:
cot(pi) = cos(pi)/sin(pi)
= -1/0
so you have a negative over 0, approaching from the -ve side of pi wouldn't it be +infinity? why is it -infinity?


additionally this confuses me because a previous question I was working went like:

1)lim x->-3+ (x+2)/(x+3) = - infinity
2)lim x->-3- (x+2)/(x+3) = + infinity
when substituting in 3, one would get a -ve int/0.
so i thought you found out whether it is +ve or -ve infinity by multiplying signs.
1) -3+ so take + times - (from -ve int) = -ve ...and you get -ve infinity
2) -3- so take - times - (from -ve int) = +ve ...and you get +ve infinity

but that was the way a friend showed me, its worked for all the questions up until the cotx one. any help in understanding is much appreciated, thanks.

 

[SammyS]

Homework Statement

lim x->pi- cot(x)

Homework Equations

cot(x) = cos(x)/sin(x)

The Attempt at a Solution

so substituting pi into:
cot(pi) = cos(pi)/sin(pi)
= -1/0
so you have a negative over 0, approaching from the -ve side of pi wouldn't it be +infinity? why is it -infinity?


additionally this confuses me because a previous question I was working went like:

1)lim x->-3+ (x+2)/(x+3) = - infinity
2)lim x->-3- (x+2)/(x+3) = + infinity
when substituting in 3, one would get a -ve int/0.
so i thought you found out whether it is +ve or -ve infinity by multiplying signs.
1) -3+ so take + times - (from -ve int) = -ve ...and you get -ve infinity
2) -3- so take - times - (from -ve int) = +ve ...and you get +ve infinity

but that was the way a friend showed me, its worked for all the questions up until the cotx one. any help in understanding is much appreciated, thanks.

What is the sign of sin(x) when x is a little less than π ?

 

[shocklightnin]

positive..?

 

[Mentallic]

Right, so it should be \frac{-1}{0^+} because we're looking at \sin(\pi ^-)

 

[shocklightnin]

ooh right right! so that's why its -ve infinity. ah thanks, got it now :P