Calculating Limits: Understanding Lim 1 - cos(x)/x^2 x->0

[Halcyon99]

Lim 1 - cos(x)/x^2
x->0

My attempt:

lim [1 - cos (x)/ x][lim 1/x] = (1)(1/0)

Please help me with some basic examples.

 

[poolwin2001]

lim [1 - cos (x)/ x]=1?
How ??
While solving limit Problems, you make use of certain solutions
like
\lim_{x\rightarrow 0} \frac{Sin(x)}{x}=1

\lim_{x\rightarrow 0} \frac{e^x-1}{x}=1

==============================================
eg:\lim_{x\rightarrow 0} \frac{Tan(x)}{x}


=>\lim_{x\rightarrow 0} \frac{\frac{Sin(x)}{cosx}}{x}


==>\lim_{x\rightarrow 0} \frac{Sin(x)}{x} \lim_{x\rightarrow 0} \frac{1}{Cosx}
==>1*1=1
end example==================
In this case,try to convert this into a known limit
HINT:cos2x=1-2sin^2x

 

[JonF]

use le'hopitals(spelling?) rule

 

[berkeman]

lim [1 - cos (x)/ x]=1?

Hmm, I got 1/2. I think you left the x^2 out of the original denominator:

lim x-->0 of ( 1-cos(x) )/x^2

= lim x-->0 ( sin(x) )/2x

= lim x-->0 ( cos(x) )/2 = 1/2


BTW, here's a good page on L'Hopital's Rule:

http://www.math.hmc.edu/calculus/tutorials/lhopital/


PS -- I went back and used a calculator to plug in small numbers for x, and it looks more like the limit of the original function of x goes to zero. I wonder if I did the first differentiation wrong, or the calculator is fooling me. What do other folks get?

 

[sushant]

Answer

I guess this can be solved in eitherof following 2 ways.:

1. Using formula of cosx=1-(x^2)/2!+(x^4)/4!...
so that
(1-cosx)=x^/2-(x^4)/4!...
=x^2[1-(x^2)/4!...]
so that
Lt x->0 of (1-cosx)/x^2 becomes :
=Lt x->0 of 1/2[1-(x^2)/4!...]
=1/2

2. other method could be usin L'Hospital's rule using differentiation
this too yields answer as 1/2

take care

 

[Muzza]

PS -- I went back and used a calculator to plug in small numbers for x, and it looks more like the limit of the original function of x goes to zero. I wonder if I did the first differentiation wrong, or the calculator is fooling me. What do other folks get?

Your calculator was probably using degrees instead of radians.

 

[berkeman]

>Your calculator was probably using degrees instead of radians.

Perfect! Thanks, Muzza. I thought I was losing my mind!

I was going to suggest to Halcyon99 that a good way to check the results you get from L'Hospital's Rule is to just plug some small numbers into the original fractional expression using a calculator. Now I know to add the caveat that the calculator needs to be in Radian mode. I'd spaced the part about when series expansions or other expressions are shown sharing a variable between trig functions and algebraic functions, the trig function arguments are in radians.

Thanks again, -Mike-