# Limits of trig functions

1. Dec 5, 2009

### mathProb

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

lim (cos x - 1) / (sin^2 x + x^3) as x approaches 0.

2. Relevant equations

sinx/x = 1

3. The attempt at a solution
I get 0/0. Is that the answer?

2. Dec 5, 2009

### Dick

Yes, it has 0/0 form. That doesn't mean the limit is 0. Do you know l'Hopital's rule?

3. Dec 5, 2009

### ideasrule

Just a note: most of the limits you do will be in the form 0/0. What the limit actually equals to can be anything from negative infinity to positive infinity.

4. Dec 6, 2009

### mathProb

I know l hospitals rule u take the derivative and then find the limit

but is there another way u can so it by using sinx/ x = 1

5. Dec 6, 2009

### Dick

Yes, there's another way. Start by multiplying numerator and denominator by cos(x)+1 and expand the numerator. Try it and see how far you get.

6. Dec 6, 2009

### mathProb

cos^2 - 1 / ( sin^2 + x^3)(cos x + 1)
= - sin^2/ (sin^2 + x^3 )(cos +1)

is this the right way?

7. Dec 6, 2009

### mathProb

i still get 0/0

8. Dec 6, 2009

### Dick

You are doing fine. Now divide numerator and denominator by sin(x)^2.

9. Dec 6, 2009

### mathProb

- 1/cos + 1 + sin^2/x^3 cos x + sin^2/X^3
limit = 1?

10. Dec 6, 2009

### Dick

You already weren't using enough parentheses to make it clear what you mean. Now you've lost all of them. I have no idea what you are trying to write. Start over and show the algebra steps you are using.

11. Dec 7, 2009

### mathProb

(1/cos) + 1 + ?(sin^2/x^3 cos) + (sin^2/x^3)

limit =1

12. Dec 7, 2009

### Schrodinger's Dog

13. Dec 7, 2009

### Dick

Start again from -sin(x)^2/((sin(x)^2+x^3)(cos(x)+1)). And explain step by step you got there. Divide numerator and denominator by sin(x)^2. There's no need to multiply the denominator out. What's lim x->0 of cos(x)+1?

14. Dec 7, 2009

### Schrodinger's Dog

Whs^

BTW I cut and pasted a png into my blog and got warned for it as a Homework Q?

Is that really necessary?

https://www.physicsforums.com/blog.php?b=1522 [Broken]

No offence but I don't think trig identities warrant that much attention do they?

Last edited by a moderator: May 4, 2017
15. Dec 7, 2009

### Dick

No idea. I can't see the blog entry anyway. You should take it up with the mentor who issued the warning. BTW, I don't see what sigmoids have to do with this problem. It's just a simple limits exercise.

Last edited by a moderator: May 4, 2017
16. Dec 7, 2009

### Schrodinger's Dog

Sure np thanks Dick, helpful as ever and quick.

And congrats on being made a mod/homeworkhelper/Mentor btw, well deserved.

17. Dec 7, 2009

### mathProb

1+ (sin^2/x^3(cosx+1))

limit of cos + 1 = 2

18. Dec 7, 2009

### Dick

Yes, the limit of cos(x)+1 is 2. As for "1+ (sin^2/x^3(cosx+1))", if that's supposed to be the denominator, it's still messed up. If you aren't going to show your algebra steps, I really can't tell you how you are messing up.

19. Dec 7, 2009

### mathProb

-sin^2 / (sin^2 + X^3(cos +1))
= 1/(1+ (x^3(cos +1))/sin^2)
lim= 1

20. Dec 7, 2009

### Dick

That's -sin(x)^2 / ( (sin(x)^2+x^3)*(cos(x)+1) ). (cos(x)+1) multiplies both sin(x)^2 and x^3, not just one of them. So you get -1/( (1+x^3/sin(x)^2)*(1+cos(x)) ). Now what's the limit of x^3/sin(x)^2?