Scientific Computing - Quiz 1 - Question 1

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.