Limits - The Gateway to Calculus
A limit asks a simple question: as an input value gets closer and closer to some number, what value does the output of a function approach? Limits are the foundation stone of all of calculus – they make derivatives and integrals possible.
When Isaac Newton and Gottfried Wilhelm Leibniz independently invented calculus in the 1660s and 1670s, they relied on the intuitive but logically shaky idea of “infinitesimals” – quantities that were treated as smaller than any real number yet somehow not quite zero. This vagueness troubled mathematicians for well over a century, most famously the philosopher George Berkeley, who mocked infinitesimals in 1734 as “the ghosts of departed quantities”. The problem was not fully resolved until the French mathematician Augustin-Louis Cauchy gave a rigorous definition of the limit in 1821, later sharpened into the precise “epsilon-delta” form still taught today by the German mathematician Karl Weierstrass in the 1860s. That formalisation finally put calculus on solid logical ground, nearly two centuries after it was first used to launch modern physics.
The Intuitive Idea
Imagine walking along a path towards a door. You get closer and closer – but the question of limits is: where does the path lead, not whether you actually walk through the door. Even if the door is locked (the function is undefined at that exact point), the limit can still exist if your path clearly heads towards a specific place.
Notation
We write a limit like this:
limx → a f(x) = L
Read as: “the limit of f(x) as x approaches a equals L.”
It means: as x gets arbitrarily close to a (from either side), f(x) gets arbitrarily close to L.
One-Sided Limits
Sometimes a function approaches different values from the left and from the right.
Left-hand limit: limx → a− f(x) – x approaches a from values less than a.
Right-hand limit: limx → a+ f(x) – x approaches a from values greater than a.
The two-sided limit exists only when both one-sided limits are equal and equal to the same value L.
When a Limit Does Not Exist
- The left and right limits are different.
- The function grows without bound (heads to ±∞).
- The function oscillates wildly near the point.
The Limit Laws
| Law | Statement |
|---|---|
| Sum | lim[f(x) + g(x)] = lim f(x) + lim g(x) |
| Difference | lim[f(x) − g(x)] = lim f(x) − lim g(x) |
| Product | lim[f(x) · g(x)] = lim f(x) · lim g(x) |
| Quotient | lim[f(x)/g(x)] = lim f(x) / lim g(x) (if lim g(x) ≠ 0) |
| Constant | lim[c · f(x)] = c · lim f(x) |
| Power | lim[f(x)]n = [lim f(x)]n |
Evaluating Limits
Direct substitution: If f is continuous at a, simply substitute x = a.
Factoring: When substitution gives 0/0, factor and cancel the common term.
Rationalising: When surds are involved, multiply by the conjugate.
L'Hopital's Rule: For 0/0 or ∞/∞ forms, differentiate numerator and denominator separately.
Worked Examples
This polynomial is continuous everywhere. Direct substitution:
3² + 2(3) − 1 = 9 + 6 − 1 = 14.
Direct substitution gives 0/0 – indeterminate form. Factor the numerator:
(x² − 4) = (x − 2)(x + 2).
Cancel (x − 2): limit becomes limx → 2 (x + 2) = 2 + 2 = 4.
Divide every term by x² (the highest power):
(3 + 5/x²) / (1 − 1/x²). As x → ∞, the fractions vanish: 3/1 = 3.
Continuity and Limits
A function f is continuous at x = a when three conditions hold:
1. f(a) is defined.
2. limx → a f(x) exists.
3. limx → a f(x) = f(a).
If any condition fails, there is a discontinuity at that point.
Common Mistakes
| Mistake | Correction |
|---|---|
| Assuming lim f(x) = f(a) always | Only true when f is continuous at a |
| Concluding no limit when f(a) is undefined | The limit can exist even if f(a) is undefined |
| Stopping at 0/0 without further work | Factor, rationalise, or use L'Hopital's Rule |
Key Takeaways
- A limit describes the value a function approaches, not necessarily the value it takes.
- The two-sided limit exists only if the left and right limits agree.
- Limit laws let you break complex limits into simpler parts.
- 0/0 and ∞/∞ are indeterminate – they require further algebraic work.