Problem:
L'Hopital's rule is a method for determining the value of indeterminate forms. Determine the value of the following limits
$ \lim\limits_{x \to 0}{\frac{\sin x}{x}} = 1 $
Solution:
It is really funny that the question gives out the answer - the problem already give us the limit!
Alright, let's derive it anyway, if we assume the derivative of sine is cosine, then that is easy. We have:
$ \begin{eqnarray*} & & \lim\limits_{x \to 0}{\frac{\sin x}{x}} \\ &=& \lim\limits_{x \to 0}{\frac{\cos x}{1}} \\ &=& 1 \end{eqnarray*} $
Note that we carefully said that we assume the derivative of sine is cosine above. It is a fact! Why do we bother to carefully assume it? The problem with the above is that it is a logically wrong proof! We actually need this limit to prove the derivative of sine and cosine, it is a circular reasoning.
The correct way of proving the identity should be found in any high school text book. This is based on the inequality $ \theta \ge \sin\theta \ge \theta\cos\theta $. The inequality can be argued geometrically, and then we divide through $ \theta $ and apply squeezing principle to get to the limit we wanted.
L'Hopital's rule is a method for determining the value of indeterminate forms. Determine the value of the following limits
$ \lim\limits_{x \to 0}{\frac{\sin x}{x}} = 1 $
Solution:
It is really funny that the question gives out the answer - the problem already give us the limit!
Alright, let's derive it anyway, if we assume the derivative of sine is cosine, then that is easy. We have:
$ \begin{eqnarray*} & & \lim\limits_{x \to 0}{\frac{\sin x}{x}} \\ &=& \lim\limits_{x \to 0}{\frac{\cos x}{1}} \\ &=& 1 \end{eqnarray*} $
Note that we carefully said that we assume the derivative of sine is cosine above. It is a fact! Why do we bother to carefully assume it? The problem with the above is that it is a logically wrong proof! We actually need this limit to prove the derivative of sine and cosine, it is a circular reasoning.
The correct way of proving the identity should be found in any high school text book. This is based on the inequality $ \theta \ge \sin\theta \ge \theta\cos\theta $. The inequality can be argued geometrically, and then we divide through $ \theta $ and apply squeezing principle to get to the limit we wanted.
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