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This might be a pretty stupid question. But why is it that while applying limits to an exponential function like- [itex] \lim_{x\rightarrow 0} e^{f(x)} [/itex] we move the limit to only the part of the expression which involves the variable on which the limit is being evaluated and hence we now write it as- [itex]e^{(\lim_{x\rightarrow 0} f(x))} [/itex]? Can't we evaluate the limit without reducing the terms inside the limit?

What I mean is earlier, the limit included [itex] e [/itex] withing itself too. But when we simplified it, the [itex] e [/itex] came out of the limit and instead the limit was being operated on only [itex] f(x) [/itex]. Can we do this? Is there any formal rule or formula for this like other formulas for evaluating and simplifying limits?

So my question is basically that is this a rule or formula which we follow or do we do it just because of logic? (The logic of applying the limit only to the part of the expression which gets affected by the limit)

What I mean is earlier, the limit included [itex] e [/itex] withing itself too. But when we simplified it, the [itex] e [/itex] came out of the limit and instead the limit was being operated on only [itex] f(x) [/itex]. Can we do this? Is there any formal rule or formula for this like other formulas for evaluating and simplifying limits?

So my question is basically that is this a rule or formula which we follow or do we do it just because of logic? (The logic of applying the limit only to the part of the expression which gets affected by the limit)

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