How can a limit replace infinitesimals when defining a derivative?

In early calculus, mathematicians like Isaac Newton and Gottfried Wilhelm Leibniz defined the derivative using infinitesimals—quantities treated as being smaller than any finite number yet not equal to zero. This approach faced significant logical criticism, most notably from George Berkeley, who famously described these values as the "ghosts of departed quantities" because they were treated as non-zero during division but vanished when convenient.123

To resolve these logical inconsistencies, 19th-century mathematicians, including Karl Weierstrass, replaced infinitesimals with the rigorous concept of a limit. Instead of relying on an infinitesimal change $dx$, the derivative is defined as the limit of the average rate of change, $\frac{\Delta y}{\Delta x}$, as $\Delta x$ approaches zero.142

In this framework, $\Delta x$ is never set to zero; rather, the definition examines the behavior of the ratio as $\Delta x$ becomes arbitrarily small. This allows mathematicians to assign a precise value to the derivative without requiring the existence of "infinitely small" numbers, effectively turning approximations into exact quantities.562