trig limits (1 Viewer)

[Owyn]

hey, everyone, i came across a maths problem and the text book i have is useless in explaining it so I was hoping someone here might be able to help.

lim x approaches 0
1-cos(2x)
x*x

 

[Estel]

lim x--> 0 ([1-cos(2x)]/x^2)
= lim x--> 0 (2sin^2x/x^2)
= 2 lim x--> 0 (sinx/x)^2
= 2.1^2
= 2
I hope this is right as I have never done trigonometric limits.

 

[Owyn]

yeh, according to the book answers that is right, thanks

 

[acmilan]

Isnt this extension 1?

 

[Estel]

Why would it be?
All it requires is knowledge that sinx is approx. equal to x for small x.

EDIT: 2U Syl ref 13.4

 

[acmilan]

but you used double angle formulae?

 

[Estel]

EDIT:
Faulty Proof Removed.

 

[CM_Tutor]

Originally posted by acmilan1987
Isnt this extension 1?

Yes, it is.

Originally posted by Estel
= lim y--> 0 (2/y^2 - 2cosy/y^2)
= lim y--> 0 (2/y^2)(1 - cos^2y)

I don't follow this step - where did the squared come from?

 

[Estel]

Trust you to find some error
I was working backwards and ahh well...

How bout this one:

lim x--> 0 (1-cos(2x)/x^2)
= lim x--> 0 (1-cos^2(2x))/[(1/4)(2x)^2(1+cos(2x))]
= lim x--> 0 4(1-1cos^2(2x))/[(2x)^2(1+cos(2x))]
= 4 lim x--> 0 sin^2(2x)/(2x)^2(1+cos(2x))
= 4 lim x--> 0 1/(1+cos(2x))
= 4 . 1/2
= 2

I think it's right... and no double angle formulae.

 

[kpq_sniper017]

Can't you just say:

cos2x=1-2sin²x
.'. 2sin²x=(1-cos2x)

.'. lim x-->0 (1-cos2x/x²)
= 2 lim x-->0 (sin²x/x²)
= 2x1
= 2

It uses double angle formulae, but it works (I think).

 

[Estel]

That's the one i used at the start... just trying (with some futility) to prove a point that you don't need the double angle formulae and hence may be appropriate for 2U.

 

[sneaky pete]

limit x -> 0

d/dx (1-cos(2x)) / d/dx (x^2)
d/dx (2sin(2x)) / d/dx (2x)
4cos(2x) / 2
4cos(2*0) / 2
4/2 = 2

 

[CM_Tutor]

Estel, I think that one is OK, but it's also well beyond what might be expected of a 2u student.

Sneaky Pete, I would recommend against using L'Hopital's Rule in the HSC - especially in 2 or 3u.

 

[sneaky pete]

'L'Hopital's Rule' , its not in 2 or 3 unit maths ??

I think they should teach it in all maths because it's so handy in first year uni maths

 

[CM_Tutor]

L'Hopital's Rule is not in ANY HSC maths course. I agree it's useful, but equally there is value in developing understanding of limits without using such approaches.

 

[kpq_sniper017]

Originally posted by Estel
That's the one i used at the start... just trying (with some futility) to prove a point that you don't need the double angle formulae and hence may be appropriate for 2U.

oh, whoops.

trig limits are part of ext 1 anyway, so a 2U student wouldn't bas asked a question like that.