"Gets close to" is motion language, and there's no motion here. A function is a static object; x isn't sliding anywhere. For 190 years everyone waved at this and it mostly worked — until it didn't, and mathematicians found functions where intuition gave flatly wrong answers.

Weierstrass's fix in the 1860s replaces motion with a challenge-and-response game:

Read it adversarially. Someone challenges you: "get f within 0.001 of L." You must produce a radius around a small enough to guarantee it. If you can meet every challenge, the limit exists.

The load-bearing detail is the leading 0 <. It explicitly excludes x = a. That single symbol is what makes limits able to talk about 0/0 at all.

One level deeper: this only works because the reals are complete — they have no gaps. A sequence of fractions can march toward √2, which is not a fraction. The real numbers were constructed (Dedekind, 1872) precisely so that anything that looks like it's approaching something actually has something to approach.

Completeness is the hidden foundation under the three big existence theorems of the course — Intermediate Value, Extreme Value, and Mean Value. All three are false over the rationals alone. BC states all three and proves none, so this sentence is the one the textbook is missing.

It means the form alone doesn't determine the answer. Consider three limits as x→0, all of the form 0/0:

Same form, three different answers. The form tells you nothing; you have to dig into the specific functions. Contrast 5/0, which is not indeterminate — it always blows up. Knowing the difference tells you whether there's work to do.

Apply the Factor Theorem to both halves of the fraction. If substituting x = a gives 0/0, then the numerator is zero at a and the denominator is zero at a. So (x − a) divides both. It's there by guarantee, not by luck. Your only job is to find it.

That's why "cancel and re-substitute" always works for rational functions, and why it's the first thing to try.

These are the limits that can't be done by algebra, because the functions involved aren't polynomials and there's no factor to cancel. Each one has to be established once, from a picture or a definition, and then it becomes a tool. Every trig and exponential derivative in the course rests on one of them.

Every step of that derivation used sector area = ½r²θ, which is only true in radians. In degrees the sector area picks up a factor of π/180, and the limit comes out to π/180 ≈ 0.01745 instead of 1.

Consequence: in degrees, d/dx[sin x] = (π/180)·cos x, and that ugly constant would propagate through every trig formula in mathematics forever. Radians are the unit that makes calculus clean — that's the entire reason they exist.

This limit is the derivative of e^x at x = 0 — write out the difference quotient with a = 0 and you get exactly this expression.

So the question "why is it 1?" is really the question "what is e?" And the answer: e is defined as the base for which the exponential curve has slope exactly 1 where it crosses the y-axis. Every exponential y = b^x passes through (0,1). They differ in how steeply. For b = 2 the slope there is about 0.693; for b = 3 it's about 1.099. Somewhere between 2 and 3 there's a base where the slope is exactly 1. That base is e ≈ 2.71828.

Everything convenient about e follows from that one choice — including d/dx[e^x] = e^x, the only function that is its own derivative.

Invest $1 at 100% annual interest. Compounded once: $2. Compounded twice a year at 50% each: (1 + ½)² = $2.25. Quarterly: (1 + ¼)⁴ ≈ $2.44. Daily: ≈ $2.7146. Continuously — the limit as n→∞ — you get e ≈ 2.71828, and no more. Compounding infinitely often doesn't give you infinite money.

Jacob Bernoulli found this in 1683 studying exactly this question. The general form with x in place of 1 is the same statement at interest rate x.

This matters later: it's why continuous growth models use e^kt, and it reappears in Unit 7 as the solution to y′ = ky.

ln x ≪ x^p: logarithms grow like the number of digits. Going from a thousand to a trillion multiplies x by a billion and raises ln x by about 21. Nothing polynomial can be beaten that slowly.

x^p ≪ e^x: differentiating a polynomial lowers the degree; do it p+1 times and it's gone. Differentiating e^x changes nothing. Apply L'Hôpital enough times and the polynomial dies while the exponential stands there.

e^x ≪ x!: e^x multiplies by a fixed e at each step; x! multiplies by an ever-growing factor. Fixed ratio always loses to growing ratio.

x! ≪ x^x: x! is x·(x−1)·(x−2)···, all factors below x; x^x is x multiplied by itself x times.

It's not a limit trick; it's the ranking of how fast things can grow, and it recurs constantly:

• In this course, Unit 10: whether an infinite series converges is entirely a question of whether its terms shrink fast enough. Every convergence test is the hierarchy in disguise.
• Improper integrals: ∫₁^∞ dx/x^p converges only when p > 1. The threshold is a growth-rate threshold.
• Computer science: an algorithm that takes ln n steps is fine at any scale; n² is usable; 2ⁿ is unusable past about n = 50; n! is unusable past about n = 15. Same ordering, and it's the reason some problems are considered intractable.
• Compound growth generally: exponential eventually beats polynomial always, however unfavourable the constants. A quantity growing 1% a year overtakes any fixed-power trend, given enough time. That's the mathematical content of most arguments about compounding.

Near the point a, each function is well approximated by its tangent line. If f(a) = g(a) = 0, those tangent lines both pass through zero there, so near a:

The ratio is then f′(a)(x−a) / g′(a)(x−a), and the (x−a) cancels. What's left is f′(a)/g′(a).

L'Hôpital is just cancelling the common factor again — the same move as every algebraic technique above, done with tangent lines instead of factoring. That's why it needs the 0/0 condition: without it, the tangent lines don't both pass through zero and the cancellation isn't available.

Historical aside worth having: L'Hôpital didn't discover it. He paid Johann Bernoulli a retainer for exclusive rights to his mathematical output, published it in the first-ever calculus textbook in 1696, and credited him only vaguely. Bernoulli complained about it for the rest of his life.

Continuity means taking the limit and evaluating the function commute — you can do them in either order and get the same thing. That's the useful form, because it's exactly what licenses "just substitute" as a limit technique, and it's what lets you pass a limit inside a continuous function later on.

Informally: a continuous curve can't get from below a line to above it without crossing it. Obvious — for the reals.

It is false over the rationals. Take f(x) = x² − 2 on [1, 2] with only rational inputs allowed. f(1) = −1, f(2) = 2, so it must cross zero — but the crossing point is √2, which isn't rational. The function jumps over zero without ever landing on it.

So IVT isn't a fact about continuity alone. It's a fact about continuity plus the completeness of the reals. This is the payoff of that earlier note about gaps, and it's the reason existence theorems in calculus are theorems rather than observations.

Weierstrass's function (1872): continuous everywhere, differentiable nowhere. A curve with no smooth point anywhere — no tangent line at any location, infinitely wrinkled at every scale.

Before it, mathematicians assumed continuity basically implied smoothness apart from isolated corners. Hermite called it "a lamentable plague." It's the ancestor of fractals and of Brownian motion — actual stock price paths and actual pollen-grain trajectories are exactly this kind of object. The intuition "continuous means you can draw it without lifting the pen, so it must have a direction almost everywhere" is false, and this is the counterexample.

Next stop — real analysis. Redo the entire year with proofs. Where ε-δ is the point, where completeness gets stated honestly, and where the pathological examples (Weierstrass's function, the Cantor set) stop being curiosities and become the objects that force the definitions.

Then — measure theory and the Lebesgue integral, which fixes the Riemann integral's limitations and is the actual foundation of modern probability. Also topology, which is what you get when you keep "nearness" and throw away distance and number entirely.

And the wildcard: nonstandard analysis (Robinson, 1966), which vindicates Leibniz by constructing infinitesimals rigorously. If she ever objects that treating dy/dx as a fraction "shouldn't be allowed," this is the field that says she was right the whole time.