Defining Limits and Using Limit Notation

A limit describes the value a function approaches as the input approaches a specific point, regardless of whether the function reaches that value.

Limits10–12% of exam

Understand It

Ace It

Context

What this topic is and why it exists

Limits are the foundation of calculus, defining what value a function approaches as the input nears a particular point.

This isn't about what the function equals at that point—it's about the behavior as you close in.

A common mistake is to conflate the value of the function at a point with the limit as it approaches that point.

They are not the same.

For example, a function might not even be defined at a point, yet it can still have a limit there.

You must distinguish between evaluating a function directly and understanding its approaching behavior.

This understanding is crucial for recognizing asymptotes and holes in graphs.

Limits are expressed using notation like lim x→c f(x) = L, where L is the value the function approaches.

Understanding this notation is non-negotiable because it forms the bedrock for differentiating and integrating functions later in the course.

Always check if a function is continuous at a point, as continuity implies the limit equals the function's value at that point.

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