flowchart TD Start(["Given lim f(x) as x→c"]):::startNode Try["Try substituting x = c into f(x)"]:::processNode Check{"Does f(c) produce
a real number?"}:::decisionNode Success(["Limit = f(c)
Direct substitution works!"]):::successNode ZeroCheck{"Is the result 0/0?"}:::decisionNode Algebraic["Use algebraic techniques
(Chapter 3)"]:::algebraNode NonzeroCheck{"Is the result nonzero/0?"}:::decisionNode Infinite["Investigate infinite limit
(Chapter 5)"]:::infiniteNode Other["Check for other
indeterminate forms"]:::otherNode Start --> Try --> Check Check -->|Yes| Success Check -->|No| ZeroCheck ZeroCheck -->|Yes| Algebraic ZeroCheck -->|No| NonzeroCheck NonzeroCheck -->|Yes| Infinite NonzeroCheck -->|No| Other classDef startNode fill:#667eea,stroke:#333,stroke-width:2px,color:#fff,font-size:16px classDef processNode fill:#a8d5ba,stroke:#333,stroke-width:2px,color:#333,font-size:16px classDef decisionNode fill:#ffd93d,stroke:#333,stroke-width:2px,color:#333,font-size:16px classDef successNode fill:#6bcb77,stroke:#333,stroke-width:2px,color:#fff,font-size:16px classDef algebraNode fill:#ff8c42,stroke:#333,stroke-width:2px,color:#fff,font-size:16px classDef infiniteNode fill:#ff6b6b,stroke:#333,stroke-width:2px,color:#fff,font-size:16px classDef otherNode fill:#c9b1ff,stroke:#333,stroke-width:2px,color:#333,font-size:16px linkStyle default stroke:#666,stroke-width:2px,font-size:14px