A couple of limit computations

[flipsvibe]

Homework Statement

Compute the limits. If they don't exist, then explain.

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

2) lim as x approaches 0 √(x^2 + x^3) / (x + 2x^2)

Homework Equations

The Attempt at a Solution

1) I replaced the sin^2x in the denominator with 1 - cos^2x and tried to work from there, but I couldn't get anything to cancel.

2) I couldn't figure out anything on this one either. I factored out an x in the denominator, but again, that didn't lead anywhere.

We are doing a semester review for the final, and I am starting to get a little nervous, because I don't remember struggling with limit computations when we learned them. Any help that can get me going the right direction would be greatly appreciated. Thank you.

Edit: This course was taught integrals first, then derivatives. We have not learned L'hospitals rule yet.

 

[physicsman2]

for the second one, did you factor out an x in the numerator?
If not, that would greatly help
knowing l'hopital's rule would also really help for the first one

 

[physicsman2]

i know you can't use l'hopital's rule, but i get -1/2 for the first one using l'hopital's rule

 

[ideasrule]

Have you learned the Taylor series? I'm guessing "no", but your course seems to be ordered in a weird way.

 

[physicsman2]

Have you learned the Taylor series? I'm guessing "no", but your course seems to be ordered in a weird way.

That's the same thing i thought. I first learned about derivatives(Calc I) then about integrals(Calc II)

 

[flipsvibe]

Thanks, I figured out the first one. I factored out an x^2 in the numerator. Giving me
√[x^2 (1+x)] / x(1+2x)
==> x √(1+x) / x(1+2x)
==> lim x --> 0 is equal to 1

 

[physicsman2]

that's what I got for the second one.
so for the first one, i have no clue how to do it without using l'hopital's rule
did you make any progress on that one?

 

[flipsvibe]

That's the way I learned them in high school: derivatives -> integrals. But we are using apostol's calculus, and he goes in the order they were discovered historically.

 

[flipsvibe]

No progress on the first one at all. I might just try to use L'hopital's rule anyways. We stopped for the semester one chapter before the book goes over it.

 

[physicsman2]

That's the way I learned them in high school: derivatives -> integrals. But we are using apostol's calculus, and he goes in the order they were discovered historically.

cool, never heard of it, but the only problem I see in the first one is that x^3.
If that wasn't there, I probably could have applied the limit in about two to three steps