We ended the preceding Chapter with a fundamental question in Calculus: how small can we make an interval $\Delta x?$ The question arose as worked to develop better and better estimates of a function’s *instantaneous* rate of change by shrinking the interval over which we calculate its average rate of change:

We can’t make $\Delta x$ equal to zero since that leads to an undefined fraction. So how small *can* we make it? This question leads to the foundational idea of “limits,” without which Calculus would not exist. In this section, we build the formal definition of a limit step-by-step, and then consider cases where the limit does, and does not, exist.