Trigonometry limit

  • Thread starter songoku
  • Start date
  • #1
1,422
67

Homework Statement


limit x to phi/2 of:
(2 + cos 2x - sin x) / (x. cos x + x. sin 2x)


Homework Equations


limit and trigonometry


The Attempt at a Solution


Using differentiation, I got zero as the answer. But I am not able to do without differentiation

(2 + cos 2x - sin x) / (x. cos x + x. sin 2x)

= (2 + 1 - 2 sin2x - sin x) / ( x cos x + 2x sin x cos x)

= (-2 sin2x - sin x + 3) / [x cos x (1 + 2 sin x)]

= [(-2 sin x - 3) ( sin x - 1)] / [x cos x (1 + 2 sin x)]

Stuck there....:mad:

Thanks
 

Answers and Replies

  • #2
tiny-tim
Science Advisor
Homework Helper
25,832
251
hisongoku! :smile:
= [(-2 sin x - 3) ( sin x - 1)] / [x cos x (1 + 2 sin x)]
very good so far :smile:

now (sinx - 1)/cosx = … ? :wink:
 
  • #3
HallsofIvy
Science Advisor
Homework Helper
41,833
956
I am going to assume you mean "pi/2".

tiny-tim's method, using a well know trig limit, works. Another way:

In the numerator it is the "sin(x)-1" that makes it 0 at pi/2. In the denominator it is cos(x).

The only way to cancel, then, is to write cos(x) as sqrt{1- sin^2(x)}= sqrt{(1- sin(x))(1- cos(x))}. Now, canceling those two gives

(2sin(x)+3)(\sqrt(1+ sin(x)))}/(x(1+ 2sin(x)))
 
  • #4
1,422
67
hisongoku! :smile:


very good so far :smile:

now (sinx - 1)/cosx = … ? :wink:
Errr..I am not very sure what you mean...(sinx - 1)/cosx = tan x - sec x ?? :confused:

Based on what HallsofIvy said, it is a well-known trig limit, but I am sorry I don't know the property. Tried to goggle for it but can't find anything. Can you help me? :D


I am going to assume you mean "pi/2".

tiny-tim's method, using a well know trig limit, works. Another way:

In the numerator it is the "sin(x)-1" that makes it 0 at pi/2. In the denominator it is cos(x).

The only way to cancel, then, is to write cos(x) as sqrt{1- sin^2(x)}= sqrt{(1- sin(x))(1- cos(x))}. Now, canceling those two gives

(2sin(x)+3)(\sqrt(1+ sin(x)))}/(x(1+ 2sin(x)))
Oops, yes I meant pi not phi.

Ah, that's the idea to eliminate cos x. I tried to figure out how to eliminate cos x but I failed. Thanks for your idea.
 
  • #5
tiny-tim
Science Advisor
Homework Helper
25,832
251
hi songoku! :smile:
Errr..I am not very sure what you mean...(sinx - 1)/cosx = tan x - sec x ?? :confused:
(sinx - 1)/cosx = (sinx - 1)(sinx + 1)/cosx(sinx + 1) = … ? :smile:

(this is basically the same as HallsofIvy's method :wink:)
 
  • #6
1,422
67
hi songoku! :smile:


(sinx - 1)/cosx = (sinx - 1)(sinx + 1)/cosx(sinx + 1) = … ? :smile:

(this is basically the same as HallsofIvy's method :wink:)
Ah I see. So many methods and I couldn't think even one...whew...

Thanks
 

Related Threads on Trigonometry limit

  • Last Post
Replies
6
Views
1K
  • Last Post
Replies
8
Views
733
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
3
Views
735
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
5
Views
870
  • Last Post
Replies
18
Views
3K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
2
Views
1K
Top