The limit, the slope, the area · three ideas, one gesture
01
The limit
What the function is about to say — whether or not it ever says it.
Walk toward $x=a$ from both sides and write down what $f$ answers. If both columns settle on the same number, that number is the limit — and it does not care what happens at $a$: the point may be missing, misplaced, or fine. The limit reads the approach, never the arrival.
The suspect
ε distance to a1
from the left—
from the right—
the limit—
f(a) itself—
The approach
ε
f(a−ε)
f(a+ε)
open circle: the value the curve refuses to take
Limit is not value$\dfrac{x^2-1}{x-1}$ has no value at $1$, yet its limit there is exactly $2$. The hole changes nothing about the approach.
02
Continuity
A function is continuous where the limit and the value agree to meet.
Here is a curve built from two pieces: $x^2$ on the left of $1$, and $x+c$ from $1$ onward. Slide $c$ and watch the seam. The function is continuous at $1$ exactly when the pieces shake hands: $\lim_{x\to 1^-}f = f(1)$. One knob, the whole definition.
The seam
c raise the right piece1
left limit at 1—
f(1), by the right piece—
the gap—
verdict—
close the gap and the two dots become one
Differentiable ⇒ continuousnever the other way: $|x|$ is continuous everywhere and still has no derivative at $0$.
03
The average rate
Average speed over a stretch, against true speed at one instant.
Between $a$ and $b$ the curve climbs $f(b)-f(a)$ over a run of $b-a$: their ratio is the average rate — the slope of the chord. The derivative at one point is what that ratio becomes when the stretch collapses. Squeeze $a$ and $b$ together and watch the two numbers argue less and less.
The stretch
a-2
b2
rise f(b)−f(a)—
run b−a—
average rate—
f′ at the midpoint—
camellia: the chord · emeraude: the tangent at the midpoint
04
The rules
Sum, product, chain — one honest curve against one tempting lie.
Take $f=\sin x$ and $g=x^2$ and differentiate their combinations. For each rule the gold curve is the true derivative, computed point by point. The camellia dashed curve is the tempting shortcut. For the sum the shortcut happens to be the truth — that is why everyone trusts it too much for the product.
The rule
largest gap, lie vs truth—
verdict—
gold: the true derivative · camellia dashed: the shortcut · emeraude: the rule, applied
(fg)′ is not f′g′watch the camellia curve run away from the gold one the moment the product is involved.
05
Variations
Where f′ is positive the curve climbs — the whole story fits in one table.
Solve $f'(x)=0$, mark the roots, read the sign of $f'$ between them: that is the tableau de variations, the most French object in mathematics. The page builds it for you — live, from the same curve you are looking at.
The curve
x₀ walk the curve0
f′(x₀)—
the curve here—
dotted camellia: f′ · gold dots: where it vanishes
06
Riemann sums
The area under a curve, paid for in rectangles — then the rectangles vanish.
Cut $[a,b]$ into $n$ strips, stand a rectangle on each, add them up. The sum is wrong — visibly wrong — and then you raise $n$ and the wrongness drains away. The number the sums are draining toward is the integral $\int_a^b f$. Nothing in analysis is more honest than watching the error die.
The bargain
a-3
b4
n rectangles8
the sum—
the integral—
error—
per rectangle—
raise n and watch the error column
Signed areabelow the axis the rectangles count negative — the integral is a balance, not a surface.
07
The fundamental theorem
Accumulate area with one hand; the other hand finds it is differentiating.
Define $A(x)$ as the signed area under $f$ from $0$ to $x$, and let $x$ slide. The marine curve that appears is $A$ — and its slope, at every single moment, equals the height of $f$. That is the whole theorem: $A'(x)=f(x)$. Integration and differentiation are the two directions of one road.
The accumulator
x₀ how far you have walked2
area so far A(x₀)—
height of f at x₀—
slope of A at x₀—
verdict—
gold: f, with the walked area shaded · marine: A, the area as a curve
Épreuves the page checks by itself—
Six commissions across the atelier — and an endless examen for the brave. The page watches, counts, and never flatters.
08
Pitfalls
The six mistakes analysis punishes without mercy.
Limit ≠ valuethe limit reads the approach. $f(a)$ may disagree, or not exist at all — the limit stands.
0/0 decides nothingit is not $1$, not $0$, not ∞. It is a question, and factoring or squeezing answers it.
Continuous ≠ differentiable$|x|$ walks through $0$ without lifting the pen and still has no slope there.
(fg)′ ≠ f′g′the product rule has a cross term for a reason. Chapter IV shows the lie diverging.
Area is signed$\int_0^{2\pi}\sin = 0$ — the two lobes cancel. Distance travelled needs $\int|f|$.
dx is not decorationit names the variable and carries the width of the strip. Lose it and substitution stops working.