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5.1 — Limits: Getting Arbitrarily Close
Around 450 BCE, Zeno of Elea asked how an arrow can ever reach its target. To get there it must first cover half the distance. Then half of what remains. Then half of that. There are infinitely many such steps, and surely infinitely many steps take infinite time — so motion is impossible.
Arrows evidently do reach targets, so something is wrong with the argument. It took two thousand years to say precisely what.
The answer became calculus, and the whole of it is two questions and one surprise. The questions are how fast is this changing and how much has accumulated. The surprise, found by Newton and Leibniz independently in the 1660s and 1670s, is that those are the same question asked backwards — so answering either one answers the other for free. Everything in this Part is built from those two questions, and the machinery starts here, with the idea Zeno was missing.
The distances are \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots and they add to
\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots = 1
Infinitely many things can add to a finite total, provided they shrink fast enough. The times shrink in the same proportion, so the total time is finite too. Zeno's mistake was assuming that infinitely many steps must take infinitely long.
Making that precise requires the idea of a limit, and the limit is the foundation on which all of calculus stands.
1. What a limit says
\lim_{x \to a} f(x) = L
Read aloud: "the limit, as x approaches a, of f of x, is L." It means: as x gets closer and closer to a, the value f(x) gets closer and closer to L.
The crucial and constantly misunderstood point: this says nothing about what happens at a. The function might be undefined there, or defined to something else entirely. The limit is about the approach, not the arrival.
Consider
f(x) = \frac{x^2 - 1}{x - 1}
At x = 1 this is \frac{0}{0}, which Chapter 1.2 showed is meaningless. The function does not exist at x = 1.
But for every other x, we can cancel:
\frac{x^2-1}{x-1} = \frac{(x-1)(x+1)}{x-1} = x+1
The cancellation is legal because x \neq 1 means x - 1 \neq 0, and dividing by a nonzero thing is fine. So as x approaches 1, the function approaches 2. Test it: at x = 1.001 the value is 2.001; at x = 0.999 it is 1.999.
\lim_{x\to 1}\frac{x^2-1}{x-1} = 2
There is a hole in the graph at x=1, and the limit tells you what value would fill it. Every derivative in Chapter 5.2 is a limit of exactly this shape — a fraction that is \frac{0}{0} at the point of interest and perfectly well-behaved around it. That is why limits had to come first.
2. Approaching from both sides
x can approach a from the left or from the right, and the two can disagree.
f(x) = \frac{|x|}{x}
For positive x this is 1; for negative x it is -1. Approaching zero from the right gives 1, from the left gives -1.
\lim_{x\to 0^+} f(x) = 1, \qquad \lim_{x\to 0^-} f(x) = -1
The little + and - mark which side. The two-sided limit exists only when both one-sided limits exist and agree. Here they do not, so \lim_{x\to0}f(x) does not exist. The graph jumps.
3. The epsilon-delta definition
"Gets closer and closer" is not a definition — it is a description with a vague word in it. Cauchy and then Weierstrass, in the nineteenth century, replaced it with something checkable. This is the moment calculus became rigorous, two centuries after it became useful.
\lim_{x\to a} f(x) = L means: for every tolerance \varepsilon \gt 0, there exists a closeness \delta \gt 0 such that whenever 0 \lt |x - a| \lt \delta, it follows that |f(x) - L| \lt \varepsilon.
Read as a challenge and a response, which is how it was designed. Someone challenges you: "make f(x) within 0.001 of L." You reply: "keep x within \delta of a, and it will be." If you can meet every challenge, however small, the limit is L.
The two Greek letters: \varepsilon (epsilon) is the tolerance on the output, \delta (delta) the closeness on the input. The condition 0 \lt |x-a| excludes x = a itself, which is the formal version of "the limit ignores the point".
The absolute values are Chapter 1.2's distance, so both conditions read as distances: "if x is within \delta of a but not equal to it, then f(x) is within \varepsilon of L."
You will not need to write these proofs. What matters is knowing that "approaches" was given a precise meaning, and that everything after this point rests on it. Before Weierstrass, calculus worked and nobody could say exactly why; after him, it had foundations. Chapter 11.5 tells that story.
4. Computing limits in practice
First, just substitute. For most functions, \lim_{x\to a}f(x) = f(a). That property is called continuity and is the subject of Section 6. \lim_{x\to 3}(x^2+1) = 10, done.
When substitution gives \frac{0}{0}, you have an indeterminate form — the answer could be anything, so you must do algebra to reveal it.
Factor and cancel, as in Section 1.
Rationalise, when there is a root:
\lim_{x\to0}\frac{\sqrt{x+4}-2}{x}
Substituting gives \frac{0}{0}. Multiply top and bottom by \sqrt{x+4}+2:
\frac{(x+4)-4}{x(\sqrt{x+4}+2)} = \frac{x}{x(\sqrt{x+4}+2)} = \frac{1}{\sqrt{x+4}+2}
Now substitute: \frac{1}{4}.
Divide by the highest power, for limits at infinity:
\lim_{x\to\infty}\frac{3x^2+5x}{2x^2-1} = \lim_{x\to\infty}\frac{3 + 5/x}{2 - 1/x^2} = \frac{3}{2}
Every term with x in the denominator vanishes as x grows. The rule this encodes: for a ratio of polynomials at infinity, only the leading terms matter. Which is exactly the reasoning behind big-O notation in Volume I, 4.1.
5. The two limits that build calculus
The derivative limit, which Chapter 5.2 develops:
\lim_{h\to0}\frac{f(x+h)-f(x)}{h}
At h=0 this is \frac{0}{0}. The whole of differential calculus is evaluating it for different functions.
The famous trigonometric limit:
\lim_{\theta\to0}\frac{\sin\theta}{\theta} = 1
Meaning that for small angles, \sin\theta \approx \theta. Check: \sin(0.01) = 0.0099998.
This is true only in radians, and it is the reason radians are the right unit (Chapter 3.3). The geometric argument: for a small angle on a unit circle, the arc length is \theta by the definition of radians, and the sine is the vertical height. As the angle shrinks the arc and the chord become indistinguishable, so the ratio approaches 1. In degrees you would get \frac{\pi}{180} instead, and that factor would then infect every derivative of every trigonometric function forever.
The small-angle approximation is used constantly in physics — the pendulum's simple harmonic motion in Chapter 6.2 depends on replacing \sin\theta with \theta, which is why a pendulum keeps good time only for small swings.
6. Continuity
A function is continuous at a when three things hold: f(a) exists, the limit exists, and they are equal.
\lim_{x\to a}f(x) = f(a)
Informally: you can draw the graph without lifting your pen. No holes, no jumps, no vertical explosions.
Two theorems about continuous functions are used constantly, and both are "obvious" facts that need completeness (Chapter 1.7) to prove.
The Intermediate Value Theorem. If f is continuous on [a,b] and f(a) and f(b) have opposite signs, then f is zero somewhere in between. A continuous path from below the axis to above it must cross it.
This is what makes root-finding possible. To solve x^3 - x - 2 = 0: at x=1 the value is -2, at x=2 it is 4. Signs differ, so a root lies between. Check the midpoint 1.5: the value is 0.875, positive, so the root is between 1 and 1.5. Repeat. This is the bisection method, and Chapter 10.2 develops it.
It is also the engine of the topological results in Chapter 3.6 — the two antipodal points with equal temperature, and the always-existing windless point, are Intermediate Value Theorem arguments.
The Extreme Value Theorem. A continuous function on a closed interval [a,b] attains a highest and a lowest value somewhere on it. Both endpoints being included matters: on the open interval (0,1) the function f(x) = x has no maximum, because whatever value you pick there is a larger one closer to 1.
This is what guarantees that the optimisation problems of Chapter 5.4 have answers to find.
7. When limits go to infinity, and when they do not exist
\lim_{x\to0}\frac{1}{x^2} = \infty
This is a statement that the function grows without bound, not that it equals some number called infinity. Infinity is not a number and cannot be reached; the notation is shorthand for "exceeds any bound you name".
\lim_{x\to0}\frac{1}{x} \text{ does not exist}
because from the right it heads to +\infty and from the left to -\infty.
\lim_{x\to\infty}\sin x \text{ does not exist}
because it oscillates forever between -1 and 1 without settling.
A vertical asymptote is where the function heads to infinity; a horizontal asymptote is a finite limit as x heads to infinity. Both are read off the same way.
Every formula above, built from scratch
None of the results in this chapter are worth memorising, because each one can be rebuilt in under a minute from something simpler. What follows is that rebuilding, one result at a time, so the formula and the reason for it sit on the same page as the explanation that needed them.
Limits
\lim_{x\to a} f(x) = L
means: you can force f(x) as close to L as anyone demands, by taking x close enough to a. Note what it does not say — nothing at all about f(a) itself. A limit describes the neighbourhood, not the point.
The algebra of limits
\lim(f\pm g) = \lim f \pm \lim g, \qquad \lim(fg) = \lim f\cdot\lim g, \qquad \lim\frac fg = \frac{\lim f}{\lim g}\ (\text{if }\lim g\ne0)
These let you evaluate almost everything by substitution. The interesting cases are the ones where substitution gives \frac00 or \frac\infty\infty, and those are exactly the cases calculus was invented for.
The three limits everything else rests on
\lim_{\theta\to0}\frac{\sin\theta}{\theta} = 1
Where it comes from — the squeeze. Draw a unit circle and a small angle \theta in radians. Compare three areas: the triangle with vertices at the centre, (1,0) and the point on the circle; the sector between those two radii; and the larger triangle out to the tangent line.
\underbrace{\tfrac12\sin\theta}_{\text{small triangle}} \;\le\; \underbrace{\tfrac12\theta}_{\text{sector}} \;\le\; \underbrace{\tfrac12\tan\theta}_{\text{big triangle}}
The sector's area is \frac12 r^2\theta with r=1, from 3.3 — circles areas volumes. Multiply everything by \frac{2}{\sin\theta}, which is positive for small positive \theta:
1 \le \frac{\theta}{\sin\theta} \le \frac{1}{\cos\theta}
Take reciprocals, which flips the inequalities:
1 \ge \frac{\sin\theta}{\theta} \ge \cos\theta
As \theta\to0, \cos\theta\to1. The middle quantity is trapped between 1 and something heading to 1, so it has nowhere to go but 1.
This limit is why radians exist. In degrees the answer would be \frac{\pi}{180}, and that factor would then infest every derivative in physics.
\lim_{h\to0}\frac{e^h-1}{h} = 1
This is the definition of e, restated: e is precisely the base for which the exponential's slope at zero is exactly 1. Every other base carries a correction factor of \ln a.
\lim_{n\to\infty}\left(1+\frac1n\right)^n = e, \qquad \lim_{x\to0}\frac{\ln(1+x)}{x} = 1
L'Hôpital's rule
\text{If } \lim\frac fg \text{ gives } \frac00 \text{ or } \frac\infty\infty, \text{ then } \lim\frac{f}{g} = \lim\frac{f'}{g'}
Why it works, in one line. Near a point where both vanish, each function is approximately its tangent line: f(x) \approx f'(a)(x-a) and g(x)\approx g'(a)(x-a). Divide, and the (x-a) cancels, leaving \frac{f'(a)}{g'(a)}.
The two ways people misuse it. It applies only to \frac00 and \frac\infty\infty; using it on \frac{2}{3} gives nonsense. And it differentiates the top and the bottom separately — it is not the quotient rule.
8. Where this shows up in your life
Every rate you have ever quoted. Speed is a limit — distance over time as the time interval shrinks to nothing. Your speedometer displays a limit.
Every "as the sample gets larger" statement in statistics. The Law of Large Numbers and the Central Limit Theorem of Chapter 7.6 are limit statements.
Every performance claim in software. "This runs in O(n\log n)" is a statement about behaviour as n approaches infinity, and the technique of Section 4 — keep only the dominant term — is exactly how big-O is computed. Volume I, 4.1.
Every numerical algorithm that iterates until convergence. Newton's method, gradient descent, and every solver in Chapter 10 produce a sequence whose limit is the answer.
Every compound interest calculation done continuously. Chapter 1.4's e was defined as a limit.
Limits are the tool. The first thing built with them is the derivative — the answer to a question people asked for two thousand years and could not formulate: how fast is something changing right now?