[FlamingCobra]

graph it on a graphing calculator.

The limit of sin(x)/x as x approaches 0 might be 1, but the function sin(x)/x is most certainly not 1.

Furthermore, I can prove it to you with SUBSTITUTION!!!!!!!!

let's use a number. Like pi.

so that's

sin(pi)
----------
pi

Using our unit circle, we find that sin(pi) = 0.

So are you trying to say that 0/3.14 = 1???

I'm sorry, but it's not. It's zero.


To whoever came up with this idea:
Your head up your butt-hole might equal one, but sine of x over x does not equal one.

DISCLAIMER:
I'm sorry if your dyson sphere fifty billion light years away got ruined because of this revelation, but I'm not liable. Therefore, I'm not paying for any damages. That is all.

PEACE!

[Polpolion]

no one claimed sin(x)/x at x = 0  to be one.

ed: Really, even with rudimentary algebra contradictions come up if you define 1/0 as 1, any other number, or even any symbol.

also your proof is erroneous

[Ghostavo]

What does the limit of sin(x)/x as x approaches 0 have to do with sin(pi)/pi?

Just for reference.

Obviously sin(x)/x is not 1 since sin is not an identity function.

[FlamingCobra]

What does the limit of sin(x)/x as x approaches 0 have to do with sin(pi)/pi?

I acknowledged that the limit of sin(x)/x as x approaches 0 is one.

However, I am challenging the assumption that sin(x)/x = 1.

I used substitution (replaced x with pi) to prove that sin(x)/x is not necessarily 1.

[Qent]

1 + 8 = 9

Using substitution, substituting -111 for 1, 1/2 for 8, and 0/0 for 9, we can show that 0/0 is defined as -110.5.

Why are you even trying to substitute pi for 0, when you can just plug it in and see that sin(0)/0 is undefined? Oh never mind. There's no possible way your calc class said that.

[Ghostavo]

No one ever said that sin(x)/x = 1 for every value of x...

That would be senseless and mean that sin is an identity function.

[FlamingCobra]

In my AP Calculus class, we have been told to replace sin(x)/x with 1 whenever we see it in an equation.

[Ghostavo]

I'm sure you heard the teacher wrong.

[Bobboau]

I am challenging the assumption that sin(x)/x = 1

who is making this assumption?

[G0atmaster]

This is the same thing as the proof that 0 = 1.  And all that one can conclude from that is that algebraically correct =/= arithmatically correct.

[Bobboau]

or a better conclusion to make would be that 'proof' was bull****, just because they go through the motions and act confident doesn't mean they are not bull****ting you.

[FlamingCobra]

The "they" is these "super smart calculus people"

My teacher looked on the internet and asked a bunch of people but nowhere could she find a proof for sin(x)/x = 1. So once when she was on a workshop trip or whatever she talked to these "super smart calculus people" and they told her that "it is just generally accepted in calculus that sin(x)/x = 1"

[Bobboau]

well its not.

[Shivan Hunter]

You're missing a huge qualifier in your statement.

It is true that sin(x)/x approximates 1 for small numbers. This is not "generally accepted", it's obvious if you look at the graph and think about it as the slope, and there are several proofs.

Sin(0)/0 makes no physical sense. It's the indeterminate form, and it's undefined (it does not represent any number). You can only work with these kinds of points using limits.

Anyone who thinks sin(x)/x = 1 for all x is a moron.

[General Battuta]

This is the same thing as the proof that 0 = 1.  And all that one can conclude from that is that algebraically correct =/= arithmatically correct.

Nope because it's not algebraically correct either

[Valathil]

No comment: http://en.wikipedia.org/wiki/Removable_singularity

[Herra Tohtori]

Yeah, a lot of assumptions like this are made in calculus, within certain terms.

Like for example at very small angles, tangent of an angle is almost the same as the sine of the angle, because the hypotenuse and the long leg of the right angle approach each other in length. It all just depends on the accuracy you want to operate at.

This does NOT mean that

tan x = sin x

but it does mean that

tan x ≈ sin x  when x→0;

this can be sometimes useful for simplifying and approximating certain types of equations and can be a valid method to use, mostly in physical applications, but I would never approve of such practice in actual mathematical exercise.

Generally it's best to write up your equation's final form, then calculate it, and round the end result up or down rather than go inserting assumptions like this around.

[Valathil]

Not an assumption its a provable property of functions and spaces

[Herra Tohtori]

Not an assumption its a provable property of functions and spaces

Only if you explicitely define the function as f(0)=1. That doesn't naturally follow from the function itself, it's just a way to fill the discontinuous gap (singularity) at zero.


Of course, intuitively it seems a natural solution for the function - but mathematically, it's not. Limit of function is not the same as function itself...


...and this thread seems to be about how FlamingCobra's maths instructor has managed to confuse the two.

[Jeff Vader]

sin x / n = sin x / n = six = 6