How do you show that $\displaystyle \displaystyle \lim_{x \to 0}\left(\dfrac{\sin{x}}{x}\right) = 1$ without drawing triangle areas?
You can use L'Hopital's Rule...
$\displaystyle \displaystyle\lim_{x\rightarrow\ 0}\left(\frac{Sinx}{x}\right)=\lim_{x\rightarrow\ 0}\left(\frac{\frac{d}{dx}Sinx}{\frac{d}{dx}x}\rig ht)=\lim_{x\rightarrow\ 0}\left(\frac{Cosx}{1}\right)$
Alternatively....
the area of the sector of a circle is
$\displaystyle \displaystyle\frac{1}{2}r^2\theta$
the area of the sector, with the segment removed is
$\displaystyle \displaystyle\frac{1}{2}r^2\,Sin\theta$
As the angle goes to zero, the segment area goes to zero.
$\displaystyle \displaystyle\frac{A_1}{A_2}=\frac{\frac{1}{2}r^2\ ,Sin\theta}{\frac{1}{2}r^2\,\theta}$
As the angle approaches zero, $\displaystyle A_1\rightarrow\ A_2$
Then as $\displaystyle \displaystyle\lim_{\theta\rightarrow\ 0}Cos\theta=1$
As $\displaystyle \theta\rightarrow\ 0$
$\displaystyle Sin\theta\ \le\theta\ \le\ Sin\theta$
No you can't. Finding the derivative of $\displaystyle \sin{x}$ involves working out $\displaystyle \lim_{h \to 0}\frac{\sin{h}}{h}$ anyway. It's an entirely circular argument.
The only way that is logically correct is to squeeze the area of a sector in a unit circle by the triangles formed by $\displaystyle \sin{x}, \cos{x}$ and $\displaystyle \tan{x}, 1$.
I disagree with Prove It that "finding the Taylor's series for sin(x) involves differentiating sin(x)". In fact, that is one way of defining sin(x).
$\displaystyle \lim_{t\to 0}\frac{sin(t)}{t}$ is a very "fundamental" limit and how you do it depends in large part upon how you define sin(t).
1) A very common way of defining f(t)= sin(t) is this: On an xy- coordinate system, draw the unit circle- the circle corresponding to $\displaystyle x^2+ y^2= 1$. Start at the point (1, 0) and measure (if t> 0) counterclockwise around the circumference of the circle a distance t (clockwise if t< 0). sin(t) is the y coordinate of the point you arrive at. You can proving that $\displaystyle \lim_{t\to 0}\frac{sin(t)}{t}= 1$ by looking at the difference between sin(t) and sin(-t) and using the area of the sector from -t to +t, sin(2t), divided by the arclength, 2t.
2) Another way of defining f(t)= sin(t) is to define $\displaystyle sin(t)= \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}t^{2n+1}= t- \frac{1}{3!}t^3+ \frac{1}{5!}t^5+\cdot\cdot\cdot\$.
With that definition we have, as Archie Meade suggested (though there is no reason to "differentiate" the sum), $\displaystyle \frac{sin(t)}{t}= \sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}t^{2n}= 1- \frac{1}{3!}t^2+ \frac{1}{5!}t^4+ \cdot\cdot\cdot$ which clearly has limit 1 as x goes to 0.
3) Yet another way is to define f(t)= sin(t) to be the solution to the "intial value problem" y"+ y= 0 with initial values y(0)= 0, y'(0)= 1. From that we can prove directly that y"(0)+ y(0)= y"(0)+ 0= 0; y'''+ y'= 0 so y'''(0)+ y'(0)= y'''(0)+ 1= 0 so y'''(0)= -1; y''''+ y''= 0 so y''''(0)+ y''(0)= 0 so y''''(0)= 0, etc. and so form the Taylor's series for sin(t).
I'm sorry but I totally disagree with (2) and (3).
$\displaystyle \sin{x}$ is NOT that polynomial by definition.
It is only written as that polynomial because you wish to find the right combination of addition, subtraction, multiplication, division and exponentiation that will enable us to evaluate $\displaystyle \sin{x}$ for certain values of $\displaystyle x$.
If $\displaystyle f(x) = \sin{x}$, the polynomial comes from
$\displaystyle f(x) = \sum_{n = 0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n$.
Notice that finding these coefficients depends upon differentiating $\displaystyle f(x) = \sin{x}$ (infinitely many times for that matter...)
So you can not define $\displaystyle \sin{x}$ to be that polynomial unless you were to find the derivative of $\displaystyle \sin{x}$, which again involves finding $\displaystyle \lim_{h \to 0}\frac{\sin{h}}{h}$ anyway.
Wherever you have to differentiate $\displaystyle \sin{x}$ in the definition, that means it must have been defined in some other way earlier.
Therefore, the only correct way to define $\displaystyle \sin{x}$ is by the unit circle definition.
L'Hopital's is the easiest way to do it, there is also the long way to do it (without geometric approximation). But again it's the long way and it's kind of pointless to use when you have l'hopital's. If you want to see it though look at this calculus resource under the pre-calculus section. It goes through the proof. Or you can always pop open a textbook, but hell if we all did that this forum probably wouldn't exist
Read the posts I have given above.
You can NOT use L'Hospital's rule to find $\displaystyle \lim_{x \to 0}\frac{\sin{x}}{x}$, because L'Hospital's Rule involves finding the derivative of $\displaystyle \sin{x}$, while finding the derivative of $\displaystyle \sin{x}$ involves finding $\displaystyle \lim_{h \to 0}\frac{\sin{h}}{h}$, the exact same limit!
Let us try to find $\displaystyle \frac{d}{dx}(\sin{x})$...
$\displaystyle \frac{d}{dx}(\sin{x}) = \lim_{h \to 0}\frac{\sin{(x+h)}-\sin{x}}{h}$
$\displaystyle = \lim_{h \to 0}\frac{\sin{x}\cos{h} + \cos{x}\sin{h} - \sin{x}}{h}$
$\displaystyle = \lim_{h \to 0}\frac{\sin{x}\cos{h} - \sin{x}}{h} + \lim_{h \to 0}\frac{\cos{x}\sin{h}}{h}$
$\displaystyle = \sin{x}\lim_{h \to 0}\left(\frac{\cos{h}- 1}{h}\right) + \cos{x}\lim_{h \to 0}\frac{\sin{h}}{h}$.
Look at that, you have to evaluate THE EXACT SAME LIMIT in order to find the derivative of $\displaystyle \sin{x}$. So you can NOT use L'Hospital's Rule. It is entirely circular reasoning.
You need to have a way to evaluate this limit WITHOUT USING DERIVATIVES! That is why you have to use the Sandwich Theorem.
However,
since the purpose of first principles is to arrive at the tangent gradient (starting with a secant),
for circular motion, we only need to take the perpendicular to the radius.
The rates of change of Sin(x) and Cos(x) can be viewed directly without reference to limits
as we consider a point rotating counterclockwise around the circumference.
If it was solely necessary to calculate the rate of change of Sin(x) using the limit of Sin(x)/x
then we would have a circular analysis.
[tex]\displaystyle \lim_{x\to{0}}\left(\frac{\sin{x}}{x}\right) = \lim_{x\to{0}}\prod_{n=1}^{\infty}\cos\left(\frac{ x}{2^n}\right)[/Math] $\displaystyle \displaystyle = \lim_{x\to{0}}\left\{\cos\left(\frac{x}{2}\right)\ cdot \cos\left(\frac{x}{2^2}\right)\cdot\cos\left(\frac {x}{2^3}\right)\cdots\cos\left(\frac{x}{2^n}\right )\right\}\right\}$.
As $\displaystyle x \to 0$, $\displaystyle \cos\left(\frac{x}{2}\right) \to 1, \cos\left(\frac{x}{2^2}\right) \to 1, \cos\left(\frac{x}{2^3}\right) \to 1, ..., \cos\left(\frac{x}{2^n}\right) \to 1. $
Therefore [tex]\displaystyle \lim_{x\to{0}}\left(\frac{\sin{x}}{x}\right) = 1.[/Math]