How to find the derivative of e^x using limits

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How to find the derivative of e^x using limits.

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So, if this is the derivation, then it will be true whatever value to put for e. But this is not true. It works only for a specific value of e.
So, where is the mistake? You said that the limit in bracket equals 1. But this is only true if the value of e is the specific value 2.718.

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Where is the proof that that expression equals 1 in the limit?

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