# Homework Help: 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.

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