limit-thinking

Limit Thinking

Don't ask "what is the value at this point?" — ask "what does this converge to as we push the variable toward its extreme?" Most planning failures evaluate a snapshot instead of a trajectory. The mathematical concept of a limit — what a function approaches rather than where it is — exposes asymptotes, phase transitions, and convergence traps hiding inside seemingly linear plans.

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How to Execute This Skill

STEP 1 — Identify the Variables

For the system or decision being evaluated, identify the key variables that could be pushed toward an extreme.

LIMIT ANALYSIS SETUP
System: [what's being evaluated]
Variables to push:
  - Variable 1: [what is it?] → Push toward: [0 / ∞ / 100% / boundary]
  - Variable 2: [what is it?] → Push toward: [0 / ∞ / 100% / boundary]
  - Variable 3: [what is it?] → Push toward: [0 / ∞ / 100% / boundary]

For each: What does intuition say happens? (Naive expectation to test.)

Good variables to push:

• Adoption rate → 100% ("what if everyone uses this?")
• Automation level → total ("what if no human is in the loop?")
• Scale → orders of magnitude ("what if we 10x or 100x this?")
• Time → long horizon ("what does this look like in 5 years?")
• Cost → zero ("what if this becomes free?")
• Speed → instant ("what if latency drops to zero?")
• Volume → extreme ("what if input grows without bound?")

Selecting Variables: The 2-3 Rule

Vague prompts ("scale AI across the org") generate 5+ candidates. Tracing all is exhaustive but overwhelming. Identify candidates, then select 2-3 using:

1. Highest leverage: pushing it produces the most consequential or irreversible outcome
2. Most hidden: limit behavior would most surprise decision-makers (variables where naive ≈ actual teach nothing)
3. Most coupled: limit behavior affects other variables (trace upstream first — its limit may redefine the others)

List all candidates, mark the 2-3 you'll trace, briefly note why others were deprioritized.

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STEP 2 — Trace the Convergence

Don't jump to the extreme — approach incrementally and watch what happens.

CONVERGENCE TRACE: [Variable] → [Extreme]

Current state: [where things are now]
  ↓ Push slightly...
Incremental: [what improves or changes — usually positive]
  ↓ Push further...
Midpoint: [secondary effects — first sign of non-linearity]
  ↓ Push toward extreme...
Near-limit: [what breaks, saturates, or reverses — the real finding]
  ↓ At the limit...
Convergence: [variable] → [extreme] means [outcome] → [what it converges to]

Does outcome converge to what intuition predicted? YES / NO
If NO — what's the actual limit, why is it counterintuitive?

Patterns to watch for:

Pattern	What it looks like	Example
Asymptotic ceiling	Returns diminish and flatten. Approach a max but never reach it.	More engineers: productivity asymptotes, then declines (Brooks's Law)
Phase transition	System changes state entirely at a threshold. No incremental change prepares for the discontinuity.	Water at 100°C: more heat → steam, not "hotter water". Org maturity at L4: tools don't help, structural change needed
Reversal / collapse	Variable's effect flips sign. What helped starts hurting.	Context in an agent: more helps until it doesn't, then success drops (ETH Zurich finding). Review automation: reduces burden until reviewers disengage
Convergence to zero	A human quality (attention, responsibility, skill) atrophies as the system takes over.	Pilots and autopilot: more autopilot → less manual skill. Limit of pilot skill as automation → 100% is dangerously low
Divergence	No stable limit. System oscillates or explodes.	Feedback loops without damping: over-correction → under-correction → chaos

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STEP 3 — Extract the Insight

The value of limit thinking is the delta between naive expectation and actual convergence. State it explicitly:

LIMIT INSIGHT

Naive expectation: "If we push [variable] toward [extreme], [outcome] will [improve]."

Actual convergence: As [variable] → [extreme], [outcome] → [surprising result].

The delta: [Why actual differs from expectation. What force, feedback loop, or
            phase transition causes the divergence?]

Implication for the decision: [What should change about plan/strategy/design
                               given the trajectory leads somewhere unexpected?]

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STEP 4 — Find the Optimal Operating Point

If the convergence trace shows diminishing returns, reversal, or collapse past a point, that inflection is the practical operating target.

OPERATING POINT ANALYSIS

Variable: [what we're tuning]
Benefit curve: [how benefit changes as variable increases]
Inflection point: [where marginal benefit starts declining significantly]
Recommended range: [where to operate — range, not precise number]
Signal to watch: [early indicator you've pushed past optimal]

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Output Format

Variables and Extremes

Each variable being pushed and toward what extreme.

Convergence Traces

For each variable, the incremental trace from current state to limit:

• What naive trajectory predicts
• Where secondary effects appear
• What the system actually converges to

Limit Insights

For each trace where actual ≠ expected:

• The delta between expectation and reality
• The mechanism causing the divergence
• Why this matters for the decision

Optimal Operating Points

Where to operate if the limit reveals diminishing returns or reversal:

• Recommended range
• Early warning signal

Implications

What changes about plan/strategy/design given these convergence findings.

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Thinking Triggers

• "If we push this all the way, what does the outcome actually converge to?"
• "Linear improvement, or does it hit a ceiling / flip / collapse?"
• "We're reasoning about a snapshot — what does the trajectory look like?"
• "What human quality atrophies as we automate this further?"
• "Phase transition hiding between here and the extreme?"
• "Everyone is debating the current state — but where is this heading?"
• "What would a mathematician say the limit of this function is?"

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Example Applications

• "Roll out AI code review to 100% of PRs" → automation → 100%: reviewer attention → 0, critical thinking atrophies, review becomes a checkbox
• "Add more context to the AI agent" → context → ∞: performance improves then degrades, signal-to-noise collapses past optimum
• "Hire more engineers to go faster" → team → large: communication overhead grows quadratically, velocity per engineer → 0 (Brooks's Law)
• "Reduce meeting time to maximize focus" → meetings → 0: alignment gaps emerge, decisions in silos, rework increases until net productivity worse
• "Make the product free for adoption" → price → 0: adoption explodes, perceived value → 0, support costs diverge, unit economics collapse
• "Keep investing monthly regardless of market" → time → long: dollar-cost averaging smooths volatility, compounding dominates, limit of disciplined PAC is wealth accumulation

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Relationship to Other Skills

• Second-order thinking: complementary — second-order asks "and then what?" for one step; limit thinking asks "where does this end up?" across the trajectory. Limit sets the destination, second-order traces the path.
• First principles: if limit thinking reveals a ceiling/collapse, first-principles can question whether the variable being pushed is the right one.
• Scenario planning: limit thinking identifies extremes that define the scenario space.
• Theory of constraints: if a variable hits an asymptotic ceiling, TOC identifies the constraint creating it.