Reading Cauchy's lecture on the derivative

[henpen]

Reading Cauchy's lecture on the derivative, I see he goes from this limit

$$\large \lim _{h \rightarrow 0} \frac{1}{ \log_A((A^h)^{\frac{1}{A^h-1}})}$$
To this one $A^h= 1+\beta$
$$\large \lim _{\beta \rightarrow 0} \frac{1}{ \log_A((1+ \beta)^{\frac{1}{\beta}})}=\frac{1}{\log_A(e)}$$

I understand the intuition behind the technique, and the result. However, is this variable-change rigorous? How can we be sure that $\beta =A^h-1$ tends to 0 in the same way as $h$ does, or do we just need to know that when $h=0 \Rightarrow \beta=0$, so the limit will be the same?

 

[arildno]

No, it isn't particularly rigorous.
Cauchy made great headway in his time to elucidate how proofs ought to be made, but it was a really tough, and interesting history how mathematicians eventually managed to make their arguments as fully rigorous as the demands of our day has become.
When it comes to the tricky business of limits, the first mathematician who systematically developed a rigorous technique for this was Karl Weierstrass, working some decades later than Cauchy.

In this case, for example, I seriously doubt that at Cauchy's time, a rigorous definition of the exponential for all real numered exponents had been made, along with proofs based on continuous variable substition and how continuity of limits with respect to kernels of functions should be proven.

That being said, Cauchy's result is true.

 

[henpen]

No, it isn't particularly rigorous.
Cauchy made great headway in his time to elucidate how proofs ought to be made, but it was a really tough, and interesting history how mathematicians eventually managed to make their arguments as fully rigorous as the demands of our day has become.
When it comes to the tricky business of limits, the first mathematician who systematically developed a rigorous technique for this was Karl Weierstrass, working some decades later than Cauchy.

In this case, for example, I seriously doubt that at Cauchy's time, a rigorous definition of the exponential for all real numered exponents had been made, along with proofs based on continuous variable substition and how continuity of limits with respect to kernels of functions should be proven.

That being said, Cauchy's result is true.

Thank you. It seems my inner pedant is becoming more mathematical.

 

[arildno]

Proofs are, really, a laborious polishing act. Geniuses like Cauchy had great intuition (that's why so much of his results are still regarded as true!), and a willingness to polish their arguments to the utmost of their ability. It is upon their shoulders we all stand, removing a bit of floss and dross that remained within their arguments..