In modern treatments of calculus, limits are used to motivate the derivation of differential calculus. However, when I searched for the history of limits I came across this Wikipedia article, which says:
...the modern idea of the limit of a function goes back to Bolzano who, in 1817...
If the modern idea of limits is dated to 1817, what kind of limits did Newton and Leibniz use if any?
Newton and Leibniz initially expressed the derivative $f'(x)$ as the ratio $\frac{df}{dx}$, where the rates of change $df$ and $dx$ are infinitesimal numbers. Analogously, the integral was defined as the sum of the ordinates for infinitesimal intervals in the abscissa. Leibniz fully embraced the use of infinitesimals, while Newton started avoiding them in his later life by forming calculations based on ratios of changes, the so-called fluxions.
Also, have a look at the Wikipedia article for calculus and the Wikipedia article for the history of calculus.
In short, calculus was introduced in terms of "infinitely small" numbers and their ratios, not in terms of modern limits. In order to understand infinitesimals better, have a look at this post: https://math.stackexchange.com/a/21209/998803 .