Limit questions

1. Oct 2, 2007

uofamath114

1. The problem statement, all variables and given/known data

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

2. Relevant equations

none

3. 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.

2. Oct 2, 2007

Dick

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

3. Oct 2, 2007

uofamath114

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

4. Oct 2, 2007

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?

5. Oct 2, 2007

uofamath114

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

6. Oct 2, 2007

Dick

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

7. Oct 2, 2007

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.

8. Oct 2, 2007

dynamicsolo

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

9. Oct 2, 2007

uofamath114

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

10. Oct 2, 2007

Dick

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

11. Oct 2, 2007

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.

12. Oct 2, 2007

Dick

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

13. Oct 2, 2007

uofamath114

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

14. Oct 2, 2007

Dick

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

15. Oct 2, 2007

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?

16. Oct 3, 2007

Dick

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?

17. Oct 3, 2007

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?

18. Oct 3, 2007

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.

19. Oct 3, 2007

uofamath114

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