Calculus I · Limits and Continuity · review

Limit Meaning Review

Section 1 Summary

Summary

• A limit describes nearby behavior, not necessarily the value at the target point. • $f(a)$ and $\lim_{x\to a}f(x)$ are separate questions. • A two-sided limit exists exactly when both one-sided limits exist and agree. • The major failure modes are a jump, unbounded behavior, oscillation, or missing domain on one side. • Tables suggest. Graphs display. Algebra confirms.

After the explanation

Use the section idea

Reading lens

What are nearby outputs doing as the input approaches the target from both sides?

Mental model

Imagine tightening a window around the target input and watching where all nearby outputs are forced to gather.

Decision

Read the left-hand and right-hand behavior separately first; combine them only after both sides approach the same output.

Common trap

The function value at the target can be missing or deliberately moved, so never substitute a plotted dot for evidence from both sides.

Check yourself

You understand the section when you can explain a limit from a graph, table, and sentence without confusing it with the function value.

Source & rights

Original instruction with traceable references.

The exposition is original. No Active Calculus exercise is reproduced verbatim. Public-domain examples were modernized and recomposed when used as inspiration.

The verified handoff declares original composition and requires owner provenance review. BetterGrades-original material remains separate from public-domain references; no source textbook PDF is published here.