There are less formal math books in Russian. My absolute favorite calculus textbook is Fikhtenholts's "A Course of Differential and Integral Calculus". It is a bit less formal than many modern texts, but somehow much more approachable.
My pet peeve about calculus books is that they almost always overlook the importance of continuity. In some extreme cases, they even start with infinitely small sequences, with some rather gnarly theorems like Bolzano–Weierstrass theorem about converging subsequences.
I think this is a mistake. It's much easier to start with continuous functions and build from there. Modern readers then can visualize the epsilon-delta formulation of limits as "zooming in" on the function. The "epsilon" is the height of the screen, and the "delta" is the "zoom level" at which the function fragment fits on the screen.
And once you "get" the idea of continuity and function limits, the other limit theorems just fall out naturally.
My pet peeve about calculus books is that they almost always overlook the importance of continuity. In some extreme cases, they even start with infinitely small sequences, with some rather gnarly theorems like Bolzano–Weierstrass theorem about converging subsequences.
I think this is a mistake. It's much easier to start with continuous functions and build from there. Modern readers then can visualize the epsilon-delta formulation of limits as "zooming in" on the function. The "epsilon" is the height of the screen, and the "delta" is the "zoom level" at which the function fragment fits on the screen.
And once you "get" the idea of continuity and function limits, the other limit theorems just fall out naturally.