The idea of a limit
A limit describes the value a function approaches as the variable gets closer and closer to a chosen number. We write limx→a f(x) to mean the value f(x) heads towards as x → a. Many limits can be found by direct substitution, but when substitution gives the form 0/0, we simplify first, usually by factorising and cancelling.
Key formula
dy/dx = limδx→0 [ f(x + δx) − f(x) ] / δx — differentiation from first principles
From chord to tangent
Take two points on a curve, P(x, f(x)) and Q(x + δx, f(x + δx)). The gradient of the chord PQ is [f(x + δx) − f(x)] / δx. As Q slides towards P, δx → 0 and the chord becomes the tangent at P. The limit of the chord gradient is therefore the gradient of the tangent, which is the derivative dy/dx.
Worked example
Evaluate limx→2 (x2 − 4)/(x − 2). Factorise: (x − 2)(x + 2)/(x − 2) = x + 2, so the limit is 2 + 2 = 4. For y = x2 by first principles, [ (x + δx)2 − x2 ] / δx = (2xδx + δx2)/δx = 2x + δx → 2x as δx → 0.
Why it matters
The derivative is defined as a limit. This is why differentiation gives the exact gradient of a curve at a point, not just an average gradient between two points.
Remember
- If substitution gives 0/0, factorise and cancel first.
- The gradient of the tangent = limit of the chord gradient.
- dy/dx is defined as a limit as δx → 0.