Limits of sin(√(1-x))/√(1-x²) and (x²+sin x)/(x²+cos x)

[skateza]

Homework Statement

lim as x approaches 1 from the left of (sin$$(\sqrt{1-x})$$)/$$\sqrt{1-x2}$$

and

lim as x approaches infinity $$(x^{2}+sinx)/(x^{2}+cosx)$$

The Attempt at a Solution

I have attempted to solve these although my brain is raw, i have done a hundred limits today because my assignment is due tomorrow and I am just lost on these ones... any starting tips

 

[Dick]

The second one is easy. sin and cos are bounded. x^2 isn't. Your first problem doesn't have enough parentheses in it to make it clear. But in any event, since it looks like it is of a 0/0 form I would use L'Hopital's rule. Is it sin of the whole thing or just of the first sqrt. And does x2 mean 2*x or x^2?

 

[skateza]

ok i fixed the parentheses,

What do u mean by sin and cos are bounded

 

[Dick]

I mean |sin(x)|<=1 and same for cos while x^2 goes to infinity. Now does x2 mean x^2 or 2*x?

 

[skateza]

sorry, yeah it means x^2

 

[Dick]

I'm still going for trying to hit the first one with L'Hopital's rule. How's the second one going?

 

[Hurkyl]

Isn't the first one just sin x / x in disguise?

 

[Dick]

Isn't the first one just sin x / x in disguise?

Sure it could be done that way. I'm still waiting for the OP to do SOMETHING. The second one is not that hard.

 

[skateza]

I've solved the second one, i got 1, easy... just divided by the highest power of x in the denom. I did it a while ago i just for got to post sorry haha... but the first one has still got me,, because their is an x^2 in the bottom, so its not quite sinx/x

 

[Dick]

You can factor (1-x^2)=(1-x)*(1+x). I was wondering if I had lost you.

 

[skateza]

perfect solved it, 1/root2

 

[Dick]

Yes you did.