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I was thinking of a $\epsilon - \delta$ Definition of Derivative at infinity (which we can take according to Can you take the derivative of a function at infinity?) of a real valued function.

I tried making one one the lines of limit, where limit at a finite point $c \in A$ of $f: A \rightarrow \mathbf{R} $ is said to be $L$ if $$ \forall \epsilon > 0 (\exists \delta(\epsilon) > 0) \text{ such that } |x - c| <\delta(\epsilon) \implies |f(x) - L| < \epsilon $$ My textbook (Bartle-Sherbert's Introduction to Real Analysis (4th Edition)) extends definition for limit at infinity as follow: A function $f: (a,\infty) \rightarrow \mathbf{R} $ is said to have limit $L$ as $ x \rightarrow \infty$ if $$ \forall \epsilon > 0 (\exists M(\epsilon) > a )\text{ such that } x >M(\epsilon) \implies |f(x) - L| < \epsilon $$ I tried doing the same for derivatives, but the problem I am facing is that definition of derivatives contain this term: $$ \left|\frac{f(x)-f(c)}{x-c} - L \right| < \epsilon $$ Where $c$ is the point where we wish to calculate, so even if I consider x above a certain values (say $x >M(\epsilon) $ I am still left with $c$.

AKP2002
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  • Infinity is not a number. The best you might do is take the derivative at x then take the limit of this expression as x tends to infinity. – Paul Jan 29 '22 at 09:26
  • See https://math.stackexchange.com/questions/2747593/can-you-take-the-derivative-of-a-function-at-infinity – Leonard Neon Jan 29 '22 at 09:37

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