Question

Hello

lim
x->pi^-

for

cot(x)

[In words,this is the limit of x as it approaches pi from the negative direction for the function cot(x). I am very confused as to how this occurs and turns out to be negative infinity. Thanks.]

Answer 1

First, we get the limit of cot(x) as x->(pi) only.
Note that cot(x) is also equal to cos(x)/sin(x). Thus,
lim cot(x) as x->pi
lim (cos(x))/(sin(x))
= cos(pi) / sin(pi)
= -1 / 0
= infinity

Now we know that it approaches infinity, but we're not sure of the sign whether it's (+) or (-). That's where the "as x->(pi)+" or "as x->(pi)-" comes in.
x->(pi)+ means that x approaches pi (=3.14159) at the right side (thus it is larger than pi, for instance 3.142)
x->(pi)- means that x approaches pi (=3.14) at the right side (thus it is larger than pi, for instance 3.140)

Note that these values are in RADIANS.
Since we need to find x->(pi)- let's use the 3.140. Just grab some calculator and substitute it:
lim (cos(x))/(sin(x)) as x -> 3.140
= cos(3.140) / sin(3.140)
= -572.96

Observe that the number is numerically large but (-) in sign. Therefore,
lim cot(x) as x->(pi)-
= (-) infinity

Hope this helps~ :3

Answer 2

*oops I made a mistake on
x->(pi)- means that x approaches pi (=3.14159) at the LEFT side (thus it is SMALLER than pi, for instance 3.140)

But the answer & other explanations are still the same. :)

Answer 3

Jai you are incredibly helpful. Thank you sir.