Rules for evaluating limits?

  • Thread starter danago
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  • #1
danago
Gold Member
1,122
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Hi. We havnt started limits in class yet, but ive come across one, which i need to evaluate.

[tex]
\mathop {\lim }\limits_{n \to \infty } 2\pi r + 2nR\sin \frac{\pi }{n}
[/tex]

For a limit like that, can i just substitute infinity into the equation? Or is there some rule against that? If so, would i be right in saying:

[tex]
\mathop {\lim }\limits_{n \to \infty } 2\pi r + 2nR\sin \frac{\pi }{n} = 2\pi r
[/tex]

Thanks for the help.
Dan.
 

Answers and Replies

  • #2
Well sir dango

What your saying to me seams perfectly correct, however, limits are not my specialties. For the limit which you defined, that is the correct answer

unique_pavadrin

EDIT: dear sir dango what i had previously sated is incorrect, as n approaches n infinity, the function appraocd 2(pi)(r+R)
 
Last edited:
  • #3
Gib Z
Homework Helper
3,346
5
[tex]
\mathop {\lim }\limits_{n \to \infty } 2\pi r + 2nR\sin \frac{\pi }{n} = 2\pi r
[/tex]

2pi r is a constant, take out of the limit. We have

[tex]
\mathop {\lim }\limits_{n \to \infty } 2nR\sin \frac{\pi }{n} = 2\pi r
[/tex]

Remember for small x, sin x=x? You can see that from the fact the tangent at x=0 is y=x (on the sine curve). So as n gets large, pi/n gets small.

The n's cancel so we get that limit is equal to 2Rpi, so the solution unique_pavadrin posted is correct.
 
  • #4
danago
Gold Member
1,122
4
[tex]
\mathop {\lim }\limits_{n \to \infty } 2\pi r + 2nR\sin \frac{\pi }{n} = 2\pi r
[/tex]

2pi r is a constant, take out of the limit. We have

[tex]
\mathop {\lim }\limits_{n \to \infty } 2nR\sin \frac{\pi }{n} = 2\pi r
[/tex]

Remember for small x, sin x=x? You can see that from the fact the tangent at x=0 is y=x (on the sine curve). So as n gets large, pi/n gets small.

The n's cancel so we get that limit is equal to 2Rpi, so the solution unique_pavadrin posted is correct.
Ahh thanks very much for that. That answer is exactly what im after, considering the context im using the equation in :biggrin:
 
  • #5
HallsofIvy
Science Advisor
Homework Helper
41,833
956
[tex]
\mathop {\lim }\limits_{n \to \infty } 2\pi r + 2nR\sin \frac{\pi }{n} = 2\pi r
[/tex]

2pi r is a constant, take out of the limit. We have

[tex]
\mathop {\lim }\limits_{n \to \infty } 2nR\sin \frac{\pi }{n} = 2\pi r
[/tex]
No, you have cancel the 2 [itex]2\pir[/itex] terms:
[tex]
\mathop {\lim }\limits_{n \to \infty } 2nR\sin \frac{\pi }{n} = 0
[/tex][/quote]
Now what you say further makes sense.

Remember for small x, sin x=x? You can see that from the fact the tangent at x=0 is y=x (on the sine curve). So as n gets large, pi/n gets small.

The n's cancel so we get that limit is equal to 2Rpi, so the solution unique_pavadrin posted is correct.
 

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