limits

Limits

When to Use

Use this skill when working on limits problems in real analysis.

Decision Tree

1. Direct Substitution
2. Try plugging in the value directly
3. If you get a determinate form, that's the answer

1. Indeterminate Form? (0/0, inf/inf)
2. Try algebraic manipulation (factor, rationalize)
3. Try L'Hopital's rule: sympy_compute.py diff on numerator/denominator

1. Squeeze Theorem
2. If bounded: find g(x) <= f(x) <= h(x) where lim g = lim h
3. Verify bounds with z3_solve.py prove

1. Epsilon-Delta Proof
2. For rigorous proof: set up |f(x) - L| < epsilon
3. Find delta in terms of epsilon
4. Verify with math_scratchpad.py verify

Tool Commands

Sympy_Limit

uv run python -m runtime.harness scripts/sympy_compute.py limit "sin(x)/x" --var x --at 0

Sympy_Diff

uv run python -m runtime.harness scripts/sympy_compute.py diff "x**2" --var x

Z3_Prove

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.