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Adit
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Adit said:View attachment 71628 In attachments, there is a pic of a page, I think there area of sector MOA should be 2π^2/x. Please tell me how this makes sense?
pat1enc3_17 said:I would solve this with L’Hospital's rule
\begin{align}
\frac{f(x)}{g(x)}:= \frac{sin(x)}{x}\qquad \text{ since}\quad g'(x)\neq 0,
\end{align}
it follows with L’Hospital's rule
\begin{align}
\lim_{x\rightarrow 0}\frac{f(x)}{g(x)}=\lim_{x\rightarrow 0}\frac{f'(x)}{g'(x)} =\lim_{x\rightarrow 0} \frac{cos(x)}{1}=1,
\end{align}
Usually, but not necessarily. There are other ways of defining "sine" and "cosine" that do not require a geometric way of getting the derivative of sin(x) and cos(x).gopher_p said:One usually uses the fact that ##\lim\limits_{x\rightarrow 0}\frac{\sin x}{x}=1## to establish ##\frac{d}{dx}\sin x=\cos x##. Using ##\frac{d}{dx}\sin x=\cos x## in the process of using l'Hopital's rule to show that ##\lim\limits_{x\rightarrow 0}\frac{\sin x}{x}=1## is a circular argument.
HallsofIvy said:Usually, but not necessarily. There are other ways of defining "sine" and "cosine" that do not require a geometric way of getting the derivative of sin(x) and cos(x).
For example, we can define sin(x) to be the power series
[tex]\sum_{n=0}^\infty \frac{x^{2n+1}}{(2n+1)!}[/tex]
and define cos(x) to be the power series
[tex]\sum_{n=0}^\infty \frac{x^{2n}}{(2n)!}[/tex]
Those series have infinite radius of convergent so converge uniformly on any finite interval and, in particular are differentiable "term by term" at any x. Differentiating the series for sin(x) term by term gives
[tex]\sum_{n= 0}^\infty\frac{(2n+1)x^{2n}}{(2n+1)!}= \sum_{n= 0}^\infty\frac{x^{2n}}{(2n)!}= cos(x)[/tex].
Another way to define sine and cosine are as solutions to an 'initial value problem':
Define cos(x) to be the function, y(x), satisfying the differential equation y''= -y with the initial conditions y(0)= 1, y'(0)= 0,
Define sin(x) to be the function, y(x), satisfying the differential equation y''= -y with the initial conditions y(0)= 0, y'(0)= 1.
By the fundamental "existence and uniqueness" theorem for differential equations, there exist unique functions satisfying those. In particular, if y is the derivative of sine, if y(x)= (sin(x))' then y'(x)= (sin(x))''= -(sin(x)) so that y''(x)= -(sin(x))'= -y. That is, y(x)= (sin(x))' satisfies the same differential equation. Further, y(0)= (sin(x))' at 0 which is 1. y'(0)= -(sin(0))= 0. That is, the derivative of sin(x) is cos(x).
The theorem of limits, also known as the limit theorem, is a fundamental concept in calculus that states that the limit of a function as its input approaches a certain value is equal to the value of the function at that point.
When asking if something is wrong in relation to the theorem of limits, it usually refers to whether or not the limit exists or if there are any errors in the calculation of the limit.
The theorem of limits is important because it is the foundation of calculus and is used in many applications, including physics, engineering, and economics. It allows us to solve problems involving rates of change, optimization, and approximation.
The theorem of limits is used in various real-life scenarios, such as determining the maximum speed of a moving object, predicting population growth, and calculating the rate of change in the stock market. It is also used in designing and testing new technologies, such as in the development of self-driving cars.
One common misconception about the theorem of limits is that it only applies to continuous functions. In reality, it can be applied to any function, whether it is continuous or not. Another misconception is that the limit of a function is always equal to the value of the function at that point, when in fact, the limit may not exist at certain points or may be different from the value of the function at that point.