Overview

This skill provides practical strategies for solving limits in real analysis. It guides you through a four-step decision tree—direct substitution, indeterminate forms, the Squeeze Theorem, and epsilon-delta proofs—and shows how to verify results with symbolic tools.

How This Skill Works

Use a decision-tree workflow: start with direct substitution, then tackle 0/0 or inf/inf with algebraic manipulation or L'Hôpital's rule, apply the Squeeze Theorem when bounds exist, or construct an epsilon-delta proof for rigor. The workflow demonstrates tool-assisted verification using Sympy and Z3 as shown in the Skill.

When to Use It

• Direct substitution yields a determinate form
• Encounter 0/0 or inf/inf indeterminate forms requiring algebraic manipulation or L'Hôpital's rule
• You need to prove a limit using the Squeeze Theorem with known bounds
• You require a rigorous epsilon-delta proof of a limit
• You want to verify limit-related computations with symbolic tooling (Sympy or Z3)

Quick Start

1. Step 1: Try direct substitution; if you get a determinate form, record the limit.
2. Step 2: If you get an indeterminate form, apply algebraic simplification or L'Hôpital's rule (`sympy_compute.py diff` on numerator/denominator).
3. Step 3: When a bound exists, use the Squeeze Theorem or craft an epsilon-delta proof and verify with the tool commands (Sympy_Limit, Sympy_Diff, Z3_Prove).

Best Practices

• Start with direct substitution and look for a determinate form
• If indeterminate, apply algebraic manipulation (factoring, rationalizing) before resorting to L'Hôpital
• Use L'Hôpital's rule only after simplifying and only on valid indeterminate forms
• Use the Squeeze Theorem when you can bound f(x) between g(x) and h(x) with known limits
• For rigor, craft an epsilon-delta proof and verify the delta-epsilon relationship with a tool like math_scratchpad.py verify

Example Use Cases

• lim_{x→0} sin(x)/x
• lim_{x→0} (1 - cos x)/x^2
• lim_{x→∞} (1 + 1/x)^x
• lim_{x→1} (x^2 - 1)/(x - 1) = 2
• Definition of derivative: lim_{x→a} (f(x) - f(a))/(x - a) = f'(a)

Frequently Asked Questions