What a limit means
The core idea: A limit asks: what value does f(x) head towards as x gets close to a number a?
It describes the behaviour of the function NEAR a point — not necessarily exactly at it.
Think of walking towards a door.
The door is the value you are heading for, even if you stop just before reaching it.
A limit is that target value.
Worked example — a smooth function
What value does f(x) = 2x + 1 head towards as x gets close to 3?
Step by step
- f(x) = 2x + 1 is a smooth straight line with no breaks, so just see what it heads to as x nears 3.
- The value it approaches is 7.
Final answer
As x gets close to 3, f(x) gets close to 7.
How limits are used in this course: In AI SL, a limit is an intuitive idea — you use it to understand the gradient of a curve and the idea of an instantaneous rate of change.
You are NOT asked to evaluate limits using algebra; that belongs to a different course.
The gradient of a curve — the problem limits solve
Curves don't have one gradient: A straight line has the same gradient everywhere.
A curve like y = x² has a different gradient at every point.
To measure the gradient near a point, we first use a straight line joining two points on the curve — a chord.
Worked example — average gradient (chord)
For f(x) = x², find the average gradient between x = 2 and x = 4.
Step by step
- Find the two y-values.
- Apply the average-gradient formula.
Final answer
The average gradient between x = 2 and x = 4 is 6.
This is only an average: The chord gradient is the average gradient over the whole interval — not yet the exact gradient at a single point.
Limits fix that.
Study smarter, not longer
Most students waste 40% of study time on topics they already know. Our AI tracks your progress and optimizes every minute.
From a chord to a tangent — the limiting process
Let the points slide together: Keep one point fixed and slide the second point towards it.
As the gap shrinks towards zero, the chord turns into the tangent, and the chord's gradient heads towards the gradient of the curve at that point.
That limiting value is the DERIVATIVE.
Worked example — gradient of y = x² at x = 2 (numerically)
Use chords with a shrinking gap h to estimate the gradient of f(x) = x² at x = 2.
Step by step
- Gradient of the chord from x = 2 to x = 2 + h is ((2+h)² − 4)/h. Try shrinking h.
- Keep shrinking h.
- The chord gradients head towards 4.
Final answer
As h → 0 the chord gradient approaches 4, so the gradient of y = x² at x = 2 is 4.
You won't evaluate this limit by algebra: In AI SL you are not asked to compute derivatives from this limit.
You use the power rule.
The limit is here to explain the IDEA behind the gradient of a curve.
Limits and rate of change
The derivative is an instantaneous rate: The same limiting process gives the instantaneous RATE OF CHANGE of a quantity. f′(x) > 0 means it is increasing, f′(x) < 0 means decreasing, and f′(x) = 0 means it is momentarily not changing.
Worked example — average vs instantaneous rate
A plant's height (cm) after t weeks is h(t) = t².
Find its average growth rate between week 2 and week 4, and compare it with the instantaneous rate at week 2.
Step by step
- Average rate = chord gradient between t = 2 and t = 4.
- Instantaneous rate at t = 2 is the gradient there (found in Section 3).
Final answer
Average rate ≈ 6 cm/week over the interval; instantaneous rate ≈ 4 cm/week exactly at week 2.
Always give units in context: When interpreting a rate of change, include units — e.g. 'the height is increasing at 4 cm per week'.
A bare number earns no interpretation credit.