Question

Find the folowing limit:
lim x->1 ((cos(x*pi/2))/(1-sqrt(x)))

Answer 1

Hint: Multiply top and bottom by \(1 - \sqrt{x}\)

Answer 2

you mean 1+sqrt(x)?
im not in school, i have test today about this and this came up. so im interested how to solve it.

Answer 3

Nope, I mean multiply top and bottom by \(1 - \sqrt{x}\). Multiplying an expression by 1 is legal. In this case, we just want to put it in a form where we can evaluate the limit right?

Answer 4

i know that. but what to do when i get result of mutiplication?

Answer 5

Because if you mulitplied \(\dfrac{\cos\left( \dfrac{\pi x}{2}\right)}{1 - \sqrt{x}} \dot\ \dfrac{1 - \sqrt{x}}{1 - \sqrt{x}}\), do you know the result of this multiplication?

Answer 6

1-2sqrt(x)+x in denominator...

Answer 7

Yes and from there, you should be able to evaluate the limit.

Answer 8

no i am not :D thats why im asking ;)

Answer 9

Hmmm. Interesting. It doesn't work that way either.

Answer 10

i tried with 1+sqrt(x) and it fails to... result is pi, but im interested in how to get to that :D

Answer 11

All you have to do is find a factor of one where upon multiplying it you get an expression in the denominator where x does not become zero when you evaluate the limit.

Answer 12

yes, but its easy to say that :D

Answer 13

Do you believe in magic? ^.^

Answer 14

no :P

Answer 15

Well you're about to :D

Don't give up on multiplying by 1+sqrt(x)/1+sqrt(x) just yet.

And you know... tell me what you get, because, well, I'm too lazy to LaTeX it out XD

Answer 16

Oh wait... you haven't LaTeX'd anything at all...
d'oy

Fine, I'll do it.

Answer 17

\[\Large \frac{(1+\sqrt x)\left[\cos\left(\frac{\pi}{2}x\right)\right]}{1-x}\]

Answer 18

i can write in latex, but im too lazy to :) i got that...

Answer 19

Okay. You can do it from here. Right?

No?

Haha... because you don't believe in magic >:)

Answer 20

i cannot. :D

Answer 21

if i decompose denominator im on start again :D

Answer 22

And because I know you're smart, I'll let you work out the details of showing why
\[\Large \cos\left[\frac\pi 2 - \frac\pi 2(1-x)\right]=\cos\left(\frac{\pi }{2}x\right)\]

(Hint: It's just algebraic simplification)

Answer 23

Now that THAT's out of the way, well, OBVIOUSLY
\[\Large \cos\left[\frac\pi 2 - \frac\pi 2(1-x)\right]=\sin\left[\frac\pi2 (1-x)\right]\]

Answer 24

ok i got it now

Answer 25

So it all boils down to this:
\[\Large \cos\left[\frac\pi2x\right]=\sin\left[\frac\pi2(1-x)\right]\]


This matters? You bet it does ^.^

Answer 26

i have sinx/x and thats trivial.... ty!!!

Answer 27

Do you believe in magic? ^.^

Maybe not.

But believe in me :D

Answer 28

:D