Find the Limit of f(x)=cscx as x Approaches x-

[uofamath114]

Homework Statement

Find the limit as x approaches x-(x from the left) if f(x) = cscx

Homework Equations

none

The Attempt at a Solution

The only way I can think of solving this is to convert it to 1/sinx which would have a limit of 0. I'm not sure if that even makes any sense though. Any help appreciated.

 

[Dick]

x approaches what? No matter what it is, the answer isn't zero.

 

[uofamath114]

Sorry the ink was smudged what looked like an x- was actually pi -(pi approaching from the left).

 

[Dick]

csc(x)=1/sin(x). If x is approaching pi the denominator is going to zero. The 'limit', such as it is, must be some kind of infinity. Which kind?

 

[uofamath114]

Since it's approaching from the left I would guess negative infinity, but I'm not 100% sure on that.

 

[Dick]

Hmm. My guess would be different. But then maybe my left is different from your left.

 

[uofamath114]

I must be confused then because the way I understood it was a limit where "x-->c-" means that we only consider values less than c. So with the limit x-->(pi)- wouldn't it have to be negative infinity since infinity would be a value greater than pi or is my logic completely wrong.

 

[dynamicsolo]

I must be confused then because the way I understood it was a limit where "x-->c-" means that we only consider values less than c. So with the limit x-->(pi)- wouldn't it have to be negative infinity since infinity would be a value greater than pi or is my logic completely wrong.

Recall that your function is 1/(sin x), so what sign does the denominator have as x approaches pi from lower values?

 

[uofamath114]

Would it be negative since it's approaching a positive value from the left?

 

[Dick]

Negative, in the non-affirmative sense. Think about it!

 

[uofamath114]

So would it be a positive value then? That would be the only other option. The reason is why it is positive is what I'm still unsure about.

 

[Dick]

What's sin(pi-0.00001). Use a calculator please if you can't draw the graph of sin.

 

[uofamath114]

I get 0.00001 when I enter sin(pi-0.00001) into my calculator.

 

[Dick]

Quite reasonable. So what's csc(pi-0.00001) and what happens as x gets even closer to pi?

 

[uofamath114]

I get 100000 and it gets progressively larger and larger the closer it comes to pi, so am I correct to assume the limit is infinity?

 

[Dick]

I get 100000 and it gets progressively larger and larger the closer it comes to pi, so am I correct to assume the limit is infinity?

It gets progressively larger and larger. I don't think you have to assume anything. It's infinity. But do you understand why? sin(pi) is zero and to the left of pi, it's positive. So?

 

[uofamath114]

csc is 1/sin, so if sin(pi) is zero a number close to sin(pi) would be a number close zero, so 1/sin(pi) would be 1/(a very small number) and would keep getting larger heading towards infinity. Am I getting close or way off again?

 

[Dick]

Yes. Except now say if the number is approaching pi- the very small number is also a very small positive number. So you can call the limit +infinity. If it's pi+ then you want to say -infinity.

 

[uofamath114]

Alright I think I understand now. Thank you for your help.