5 · IB Maths
Limits
Creep towards the forbidden point from both sides. Sometimes the two approaches agree on a value the function never takes; sometimes they refuse to meet at all.
Predict first
Start from the default function (x² − 1)/(x − 1) at x = 1. As you move the Distance δ slider towards its smallest value, what do f(a − δ) and f(a + δ) approach?
For x ≠ 1, (x² − 1)/(x − 1) = x + 1. At x = 1 ± δ the value is 2 ± δ, so both sides approach 2 as δ → 0, although f(1) itself is undefined.
Check it in the model: Drag the Distance δ slider from its default (0.794) to its left end (0.001) and read From the left and From the right (1.999 and 2.001).
from the left, x = a − δ from the right, x = a + δ the function
means f(a − δ) and f(a + δ) both get
closer to L as δ → 0.
The value f(a) itself does not matter.
| δ | f(a − δ) | f(a + δ) |
|---|
Try this
- Choose (x² − 1)/(x − 1). Slide δ from about 1 down to 0.001. What value do f(a − δ) and f(a + δ) approach, and is f(1) defined?
- Choose |x| / x. Reduce δ as far as you like. Why do the two readouts stay at −1 and +1 instead of closing in on each other?
- Choose 1/(x − 1)². Compare the table rows for δ = 0.1, 0.01 and 0.001. What happens each time δ is divided by 10?
Common error: assuming the limit is the value of the function at the point. The limit describes what f(x) approaches as x gets close to a, so it can exist (a hole) even when f(a) is undefined, and it can fail to exist even when the function is defined nearby on both sides (a jump).
More on 5: Chain and product rules · Second derivative and points of inflexion · Trapezoidal rule and Riemann sums · Displacement and distance travelled · Euler's method and differential equations · Implicit differentiation · Related rates of change · Area under a curve · Calculus of motion · Maclaurin series · Newton–Raphson method (enrichment) · Optimisation · Slope fields · Tangents and the derivative · Volume of revolution