SL 5.1—Limits and derivative concept

A limit describes the value that $f(x)$ approaches as $x$ approaches a point. At SL, estimate it from a graph or from table values on both sides; formal analytic limit calculations are not required.

The derivative is the gradient function and an instantaneous rate of change. Notation identifies the changing quantities: $f'(x)$ or $dy/dx$ for a function, $dV/dr$ for volume changing with radius, and $ds/dt$ for displacement changing with time.

Example

If table values of $f(x)$ are 3.98 at $x=1.99$ and 4.02 at $x=2.01$, they support $\lim_{x\to2}f(x)\approx4$. If the tangent gradient there is 5 metres per second, the derivative gives the local rate and its units.

The function value and limit need not agree when there is a hole, and one-sided behaviour may disagree at a jump. Do not substitute $x=2$ blindly; inspect values approaching from both sides.