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It confuses me that when finding a limit, the same function which may yield a zero in its denominator can go on to yield a real limit via a different approach. Is it not strange that the same function can resolve to two different outcomes? For example the following function: $$ \frac{dy}{d\theta}=\lim_{\Delta\theta\to0} \dfrac{sin(\theta+\Delta\theta)-sin(\theta)}{\Delta\theta} $$ I understand if the function does not exist at the limit value then getting a divide by zero makes sense. How come in this case various trig substitutions get around the issue?

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    Evaluating a limit by simple substitution is only allowed when the function is continuous. I am not sure what "two different outcomes" you are referring to. If it exists, a limit can only be one value. – 2017-02-23
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    Since the derivative of the sine function is the cosine function, the limit of the above expression for $\theta=2\pi$ would be 1 when $\Delta\theta$ approaches $0$. I don't understand conceptually how using the trig identity for the sum of two angles gets around the divide by zero problem - since the function does not exist at that point. – 2017-02-23
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    It's because the limit depends on values very close to but never actually equal to zero. – 2017-02-23
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    It is the same manipulation you would have for $$ \lim_{\Delta\theta\to1} \dfrac{\Delta\theta^2-1}{\Delta\theta-1} $$. Do you understand his this works? – 2017-02-23

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Understand the concept of limits intuitively, a limit is not a value at that point. It is the value being approached by the function hence having an undefined point within a function is possible.