AP Calculus AB

## From average change to change at an instant The average rate of change of a function $f$ over an interval from $x = a$ to $x = b$ is $$ \frac{f(b) - f(a)}{b - a}, $$ the slope of the secant line through the two points on the graph. It divides the change in $f$ by the change in $x$, so it is undefined when the two values of $x$ coincide, because the denominator would be zero. That is the difficulty with asking how fast something is changing at a single instant: there is no interval, and the quotient becomes $\frac{0}{0}$. The way round it is to shrink the interval. If $f(t)$ is the distance a car has travelled at time $t$, the average speed over $t = 2$ to $t = 2.1$, then over $t = 2$ to $t = 2.01$, then $t = 2$ to $t = 2.001$, forms a list of numbers that settles towards one value. That value is the speed at the instant $t = 2$. A **limit** is the tool that makes this precise: it describes the value a quantity is approaching, without requiring that the quantity ever reaches it. ## Definition and notation Given a function $f$ and a number $c$, the limit of $f(x)$ as $x$ approaches $c$ is a real number $L$ if $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to $c$, but not equal to $c$. It is written $$ \lim_{x \to c} f(x) = L. $$ The words "but not equal to $c$" matter. The limit records what happens near $c$ and says nothing on its own about what happens at $c$. The function might be undefined at $c$, might have a value that differs from $L$…