<s>[INST] Is there a reason why this technique is invalid?  What is $ \lim_{x \rightarrow 0} \frac{1 - \cos x}{x}$? A simple way to evaluate this limit is to substitute $0$ for $x$ in the numerator to obtain
$ \displaystyle \lim_{x \rightarrow 0} \frac{1 - 1}{x} = \lim_{x \rightarrow 0} ( \frac{1}{x} - \frac{1}{x} )
= \lim_{x \rightarrow 0} (0) = 0 $
since $ \frac{1}{x} - \frac{1}{x} = 0$ since one quantity subtracted from the same quantity is 0. This technique circumvents the problem of division by zero while utilizing the fact that $\cos(0)$ is known.
 [/INST] A counterexample:
$$\lim_{x\to 0}\frac{1-\cos x} {x^2}=\frac12,\quad\enspace\text{not }0.$$
Indeed $\;1-\cos x=2\sin^2\tfrac x2$, so
$$\frac{1-\cos x} {x^2}= \frac{2\sin^2\frac x2}{4\bigl(\frac x2\bigr)^2}=\frac12\biggl(\underbrace{\frac{\sin\frac x2}{\frac x2}}_{\underset{\textstyle 1}{\downarrow}}\biggr)^2$$
</s>