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A. The Concept of Limits

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:

\[\text{average rate of change}_{[x_1,\, x_2]} = \text{slope of line segment} = \frac{\color{purple}{\Delta y}}{\color{green}{\Delta x}} = \frac{\color{purple}{f(x_2)- f(x_1)}}{\color{green}{x_2\, -\, x_1}}\]

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.