Limit Failure Cases

Definition

Scenarios where the limit of a function does not exist (DNE) as $x$ approaches a point $c$.

How to read: “as x approaches c” Meaning: The behavior of a function’s values as the independent variable x gets arbitrarily close to a constant c.

Why It Matters

Identifying why a limit fails is critical for understanding continuity and differentiability of functions.

Core Concepts

• Jump Discontinuity: Left-hand and right-hand limits are different.
• Infinite Oscillation: Function oscillates infinitely fast near the point.
• Unbounded Growth: Function goes to $\pm\infty$ at the point.

Connected Concepts

Precise Definition of a Limit, Vertical Asymptotes