# Can you Call this a standard limit?

## Main Question or Discussion Point

lim x->o sin(nx)/x = n for n E real numbers

i havent seen it in any books but it works.

## Answers and Replies

jbunniii
Homework Helper
Gold Member
If we put ##y = nx##, then the limit is equivalent to
$$\lim_{y \rightarrow 0} \left( \frac{\sin(y)}{y/n}\right) = \lim_{y \rightarrow 0}\left( n \frac{\sin(y)}{y}\right) = n \left(\lim_{y \rightarrow 0} \frac{\sin(y)}{y}\right)$$
The limit inside the parentheses on the right hand side is standard, equal to ##1##. (Proof?)

1 person
Mark44
Mentor
lim x->o sin(nx)/x = n for n E real numbers

i havent seen it in any books but it works.
It can be derived from this limit:
$$\lim_{x \to 0}\frac {sin(x)}{x} = 1$$

$$\lim_{x \to 0}\frac{sin(nx)}{x} = \lim_{x \to 0}\frac{n \cdot sin(nx)}{nx}$$
$$= n \lim_{u \to 0}\frac{sin(u)}{u} = n\cdot 1$$

1 person
so im a genuise right for inventing it?

arildno
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Dearly Missed
so im a genuise right for inventing it?
Usually, potential candidates for the Fields Medal must wait to see if it is awarded to them.

1 person
Mark44
Mentor
so im a genuise right for inventing it?
I've seen this problem in several calculus books, so I think you're premature in saying that you invented it.

1 person
I've seen this problem in several calculus books, so I think you're premature in saying that you invented it.
problem yes, but general standard solution no. they just say lim x->o sint/t = 1
my way is far more efficient.

Mark44
Mentor
Since it has appeared as a problem in many textbooks, thousands upon thousands of students have worked it, so you can't say you invented something that many people have done before you thought of doing it.

Since it has appeared as a problem in many textbooks, thousands upon thousands of students have worked it, so you can't say you invented something that many people have done before you thought of doing it.
many people have done many things but since i have not seen anywhere

lim x->o sin(nx)/x = n

you can say that I invented it. its like you know dancing many people could have foxtrotted but its the person that calls it foxtrot and creates a perimeter of foxtrot he/she is the inventor.

The GREAT thing about my standard limit is that its FAR superior to the wellknown standard limit that is lim x->0 sinx/x = 1 this is far inferior to my limit.

arildno
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It is not "your" limit. It is a completely trivial corollary.

It is not "your" limit. It is a completely trivial corollary.
show me where lim x->0 sin(nx)/x = n is written?

i feel alot of hostility here i dont know why, i thought this forum was about comradery and discussing ideas about science without persecution.

Office_Shredder
Staff Emeritus
Gold Member
Zero, just because you write something down which has not been literally word for word written down doesn't mean it is a great discovery that you have made. Anybody who has a solid understanding of calculus would have been able to calculate your limit immediately upon seeing it.

Furthermore, not that it's terribly relevant, but your result is stated in this here yahoo answer:
lim x->0 sin(nx)/nx = 1
A genuine mathematical discovery is when you can answer a problem that people have been unable to answer before; answering a problem that people simply haven't bothered to write down before isn't a discovery, it's just applying some math to a problem.

Mark44
Mentor
We are disputing your claim that you "discovered" something that has appeared in print for many years. I don't have any calculus textbooks with me right now, but I'll bet you could find this in many of the books that are used in calculus courses, such as Thomas-Finney, Larson, and others.

Ok we have a lot of haters here today.

All im saying is that the standard limit: lim x->0 sin(nx)/x = n is FAAAAAAR superior than lim x->0 sin(x)/x = 1
its not trivial at all. the yahoo link provided above is not my Standard Limit and its not as simple and elegant.

I think people here should be more open minded about discoveries you know alot of scientist work so far they become nearsighted to great solutions and things do slip the system.

Mark44
Mentor
Ok we have a lot of haters here today.
You are confused if you think what has transpired here falls under the category of "hate." No one has said anything of a personal nature about you. We have made valid points that contradict your claim of inventing something.
All im saying is that the standard limit: lim x->0 sin(nx)/x = n is FAAAAAAR superior than lim x->0 sin(x)/x = 1
its not trivial at all. the yahoo link provided above is not my Standard Limit and its not as simple and elegant.
As already stated, "your" limit is well known, and is an easy corollary of the sin(x)/x limit.
I think people here should be more open minded about discoveries you know alot of scientist work so far they become nearsighted to great solutions and things do slip the system.
Since you don't have any more to offer, I am closing this thread.