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

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’.

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.

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!