Lesson 2 of 18Article18 minFREE
Limits: Getting Arbitrarily Close
limₓ→ₐ f(x) = L means: as x approaches a (from values near a, not necessarily equal to a), the outputs f(x) approach L. The function value at a itself can be missing or different — the limit cares about the neighbourhood.
The limit idea
limₓ→ₐ f(x) = L means: as x approaches a (from values near a, not necessarily equal to a), the outputs f(x) approach L. The function value at a itself can be missing or different — the limit cares about the neighbourhood.
Algebraic limit toolkit
- Direct substitution when f is continuous at a (polynomials, rational with non-zero denominator).
- Factor and cancel removable holes: (x²−1)/(x−1) → x+1 for x ≠ 1.
- Rationalise or multiply by conjugates for root expressions.
- For ∞ forms, divide by the highest power of x in the denominator.
Worked example
limₓ→0 (√(1+x) − 1)/x. Multiply by conjugate: [((1+x)−1)/(x(√(1+x)+1))] = 1/(√(1+x)+1) → 1/2 as x → 0.