I don't understand the rewriting that's being done in this limit:
$$\lim_{x\to0} \frac{1−\cos x}{\sin x} = \lim_{x\to0} \frac{\sin x}{\cos x} $$
Why doesn't this simplify to $\frac{\sin x}{\sin x}$?
Answer
It doesn't. You can use l'Hospital to get
$$\lim_{x\to0} \frac{1-\cos x}{\sin x} = \lim_{x\to0} \frac{\sin x}{\cos x} = \tan 0 = 0$$
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