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# Compute derivatives of a function given a limit equation that it satisfies

Consider a function that satisfies the limit relation and has a continuous third derivative everywhere. Determine the values We do some simplification to the expression first. Now, we apply the hint (that if then ), So as we have But then since has three derivatives at 0 we know its Taylor expansion at 0 is unique and is given by (Theorem 7.1, page 274 of Apostol) Hence, equating the coefficients of like powers of we have Next, to compute the limit we write Then, using the Taylor expansion of (page 287 of Apostol) we know as we have Therefore we have 1. tom says:

Would I be correct in assuming the motivation for taking logs, for the limit in the second part, is because 1+f(x)/x equaled the taylor series for e^(2x+o(x)) ? Just making sure I’m not missing a ‘trick’.

• RoRi says:

Yeah, I think I did that to avoid dealing with the exponent until the end, since that was the trickiest part in the limit. This whole thing could probably be done without taking logs, but I think it gets sort of messy.

• tom says:

Very helpful; thanks. Your solution not only avoids a ‘messy’ (albiet probably interesting) alternative but also remains in the context of taylor series and o-notation. I’m glad so many examples of this type are supplied- better tools probably exist for finding limits, but these really are slightly elegant!