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walking

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The lesson I took from this was that when dealing with limits, one should not take limits for one thing before taking limits for another thing: for example, in the semicircle problem, I took the limit of each sector as the angle tends to 0 (resulting in a straight line with 0 area), before taking the limit of the summation of all the sectors in the semicircle.

Is this a general theorem when dealing with limits, and if so, what is the theorem which deals with this? If there is not a specific theorem, what is the reasoning behind it? I am aware that it may involve analysis (which I haven't studied yet), but I simply want to know if there is a rigorous reasoning which proves this.