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 2, x2-4x-2=x+2. Therefore lim_x 2 x2-4x-2=4, although the original fraction is undefined at x=2. **Input** | **1.9** | **1.99** | **2.01** | **2.1** x+2 | 3.9 | 3.99 | 4.01 | 4.1 The derivative f (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=t2, average velocity from t=2 to t=2+h is 4+h, approaching 4 as h 0. The instantaneous velocity at t=2 is 4 units per unit time.

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.

Limits and Continuity

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Limits and Instantaneous Rate

Limits and Instantaneous Rate