We can alternatively write the definition of derivative at $x_0$ by $\lim_{h \to 0}\frac{f(x_0+h)-f(x_0)}{h}$. Can we say that if $h=x/M$, $\lim_{M \to \infty}\frac{f(x_0+x/M)-f(x_0)}{x/M}$? Assume $f$ is differentiable.
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Yes, although it would make more sense to replace "$x$" with something like 1 since it's just representing some constant.
(That's assuming we're taking a limit of a function here and not taking $M$ to be an integer and taking the limit of a sequence.)
Daniel McLaury
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It definitely makes sense if $x$ is constant, but does it still make sense if $x$ is a variable? – aaka Oct 18 '14 at 23:31
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Assuming that $x$ is nonzero and doesn't vary with $M$, sure. Though it's not clear what that gains. – Daniel McLaury Oct 18 '14 at 23:33