Day 22: The Asymptotic Boundaries
Act IV: The Final Takeover
Location: Hamburg University of Applied Sciences - Advanced Mathematics Lab
It’s December 22nd. Christmas Eve is two days away. The transformation of Hamburg is nearly complete—markets open, lights glowing, mechanical systems operational.
The Grinch has invited you to your own university’s advanced mathematics lab. “Fitting,” he says, “that we should end where you study.”
On the lab’s main screen, complex graphs display rational functions—curves with asymptotes, holes, and discontinuities.
“Rational functions. The mathematics of boundaries and limits.
These describe systems with constraints—asymptotic limits that you can approach but never cross. Safety boundaries. Operational constraints. Mathematical walls.
I built my final control systems using rational functions deliberately. They create invisible barriers. Too close to an asymptote, and behavior becomes unstable. Cross a discontinuity, and the system crashes.
Understanding these boundaries is essential for maintaining systems safely. It’s engineering reality, not abstract mathematics.
And honestly? I’ve learned something about boundaries myself recently. About lines that shouldn’t be crossed. About limits that exist for good reasons.”
The Challenge
Four rational function problems protecting Hamburg’s final automated systems.
System 1: “The Vertical Boundary”
The main automation system has operational limits described by:
\[f(x) = \frac{2x + 1}{x - 3}\]
Grinch’s Note: “Every rational function has forbidden zones. Places it cannot go. Much like people, really. Find where this one fails.”
What is the vertical asymptote?
Enter the vertical asymptote (x = ?):
System 2: “Hole or Asymptote?”
A secondary control system has the function:
\[g(x) = \frac{x^2 - 4}{x - 2}\]
Grinch’s Note: “Not all discontinuities are created equal. Some are catastrophic walls. Others are merely… absences. Gaps that could be filled if someone bothered. Which kind is this?”
At \(x = 2\), is there a hole or a vertical asymptote?
System 3: “The Horizontal Limit”
The Grinch’s original cost analysis used:
\[AC(x) = \frac{500 + 10x}{x}\]
where \(AC\) is average cost per household and \(x\) is number of households.
Grinch’s Note: “What happens when you scale to infinity? The fixed costs become irrelevant. The marginal becomes everything. I wish I’d understood that sooner.”
As \(x \to \infty\), what value does \(AC(x)\) approach?
System 4: “The Hidden Equivalence”
Two control systems must synchronize:
\[h(x) = \frac{x^2 + x - 6}{x + 3}\]
and the linear function \(y = x - 2\).
Grinch’s Note: “Factor the numerator of \(h(x)\). You’ll find \((x+3)(x-2)\). Then simplify. What do you notice about the relationship between these functions?”
After simplification, how do \(h(x)\) and \(y = x - 2\) compare?
Status Update
You submit the rational function analysis:
- Vertical asymptote: \(x = 3\) (forbidden operating point) ✓
- At \(x = 2\): Hole, not asymptote (factor cancels) ✓
- Long-term average cost: Approaches €10 (economies of scale) ✓
- Function comparison: Identical except for hole at \(x = -3\) ✓
The final automated control systems unlock. Hamburg’s complete technical infrastructure is now under unified, rational control—with proper safety boundaries, understood discontinuities, and optimized scaling.
The Grinch stands and looks out the window at Hamburg, glowing with Christmas lights:
#final-technical-barrier
“That’s it. The technical barriers are gone. Everything is unlocked, everything is operational.
You’ve solved every equation. Cracked every system. Proven every economic model wrong.
Tomorrow is December 23rd. Christmas Eve is the day after.
I have one final challenge—not a barrier, but a synthesis. Day 23 will unlock the monitoring systems with logarithms. And Day 24 will combine everything you’ve learned.
It’s the difference between passing an exam and mastering a subject.
Come to the Rathaus tomorrow morning. Let’s end this the right way.”
– G. Rinch
Act IV Progress:
- ✓ Polynomial defenses cracked
- ✓ Power function grid restored
- ✓ Exponential crisis resolved
- ✓ Trigonometric systems operational
- ✓ Rational boundaries defined
- Logarithmic systems: Tomorrow
- Master synthesis: Day 24
All technical infrastructure: OPERATIONAL
Time Remaining: 2 days until Christmas Eve