Find an Approach Value across a Removable Hole
A rational expression may simplify for every nearby input except its missing point.
About 25 minutes, plus any extended practice
Open lesson 1Introductory Calculus: Limits and Continuity
A limit describes nearby approach behavior rather than necessarily a function’s value at the point. One-sided agreement, removable holes and continuity connect algebraic simplification with precise statements about a function’s local behavior.
Before you begin
- Factor quadratics; evaluate rational functions; distinguish excluded inputs from nearby allowed inputs.
Choose by skill rather than age alone. Read with an adult when useful, revisit a lesson, or explore another grade.
Follow the lesson sequence
Require Left and Right Approaches to Agree
A two-sided limit exists only when both one-sided limits exist and agree.
About 25 minutes, plus any extended practice
Open lesson 2Choose a Value that Matches the Limit
Continuity at a point requires the function value to be defined, the limit to exist, and that value to equal the limit.
About 25 minutes, plus any extended practice
Open lesson 3Compare Leading Linear Terms at Large Inputs
Dividing numerator and denominator by x reveals reciprocal terms that approach zero as x grows without bound.
About 25 minutes, plus any extended practice
Open lesson 4Review and transfer
After the lessons, try 4 new review cases and a practical task. Each answer has an explanation and a link back to the related lesson.
Open the unit reviewGuest answers are temporary. Keep your own paper or document; this unit does not certify completion or record a school grade.
Topic references
- OpenStax mathematical topic reference: topic reference — Introductory Calculus: Limits and Continuity
These references informed the topic selection. They do not certify complete standards alignment.