limits

# Limits ## When to Use Use this skill when working on limits problems in real analysis. ## Decision Tree 1. **Direct Substitution** - Try plugging in the value directly - If you get a determinate form, that's the answer 2. **Indeterminate Form? (0/0, inf/inf)** - Try algebraic manipulation (factor, rationalize) - Try L'Hopital's rule: `sympy_compute.py diff` on numerator/denominator 3. **Squeeze Theorem** - If bounded: find g(x) <= f(x) <= h(x) where lim g = lim h - Verify bounds with `z3_solve.py prove` 4. **Epsilon-Delta Proof** - For rigorous proof: set up |f(x) - L| < epsilon - Find delta in terms of epsilon - Verify with `math_scratchpad.py verify` ## Tool Commands ### Sympy_Limit ```bash uv run python -m runtime.harness scripts/sympy_compute.py limit "sin(x)/x" --var x --at 0 ``` ### Sympy_Diff ```bash uv run python -m runtime.harness scripts/sympy_compute.py diff "x**2" --var x ``` ### Z3_Prove ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "limit_bound" --vars x ``` ## Cognitive Tools Reference See `.claude/skills/math-mode/SKILL.md` for full tool documentation.