Limits and Instantaneous Rate

A limit describes the value approached as an input approaches a number. The value at that input may be different or undefined. A table or graph helps estimate a limit.

For \(x\ne2\), \(\dfrac{x^2-4}{x-2}=x+2\). Therefore \(\lim_{x\to2}\dfrac{x^2-4}{x-2}=4\), although the original fraction is undefined at \(x=2\).

The derivative \(\mathrm{f}^{\prime}(a)\) is the tangent gradient and the instantaneous rate of change at \(x=a\). A secant gives the average rate over an interval; bringing its second point towards the first estimates the instantaneous rate.

For \(s=t^2\), average velocity from \(t=2\) to \(t=2+h\) is \(4+h\), approaching \(4\) as \(h\to0\). The instantaneous velocity at \(t=2\) is \(4\) units per unit time.