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

1. Indeterminate Form? (0/0, inf/inf)

• Try algebraic manipulation (factor, rationalize)
• Try L'Hopital's rule: sympy_compute.py diff on numerator/denominator

1. Squeeze Theorem

• If bounded: find g(x) <= f(x) <= h(x) where lim g = lim h
• Verify bounds with z3_solve.py prove

1. 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.