Limits and continuity
What students will learn
Connect approaching values across representations and justify continuity, asymptotes, and intermediate outputs.
Before this topic
- Function domains and graph reading
- Algebraic and trigonometric identities
Read and connect
Read: Defining Limits and Using Limit Notation →
Distinguish a nearby limit from the value at the target, using one-sided and two-sided notation.
Read: Estimating Limit Values from Graphs →
Read nearby graph behavior, holes, jumps, oscillation, and unbounded output without mistaking a picture for proof.
Read: Estimating Limit Values from Tables →
Estimate a limit from both sides of a table and explain what finite samples cannot establish.
Read: Determining Limits Using Algebraic Properties of Limits →
Apply limit laws and composition only when their denominator and continuity conditions hold.
Read: Determining Limits Using Algebraic Manipulation →
Use factors, conjugates, and trigonometric identities to preserve nearby values while resolving a limit.
Read: Selecting Procedures for Determining Limits →
Select a limit procedure from the expression and direction of approach before calculating.
Read: Determining Limits Using the Squeeze Theorem →
Justify a trapping inequality and matching outer limits in a squeeze argument.
Read: Connecting Multiple Representations of Limits →
Translate one limit among a formula, graph, table, and verbal statement with consistent units.
Read: Exploring Types of Discontinuities →
Identify the actual obstruction to continuity and decide whether one replacement value can repair it.
Read: Defining Continuity at a Point →
Check that the function value, finite limit, and equality of the two all hold at a point.
Read: Confirming Continuity over an Interval →
Use the domain and one-sided endpoint conditions to state intervals of continuity.
Read: Removing Discontinuities →
Repair a removable discontinuity or choose a piecewise parameter by matching both sides and the target value.
Read: Connecting Infinite Limits and Vertical Asymptotes →
Use one-sided sign analysis to distinguish vertical asymptotes from removable holes.
Read: Connecting Limits at Infinity and Horizontal Asymptotes →
Compare end behavior and relative growth using justified limits, rather than a finite plotting window.
Read: Working with the Intermediate Value Theorem (IVT) →
Use continuity and bracketing to justify an intermediate output, without claiming a unique solution.
Connect: Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms →
Later connection: return to indeterminate quotient limits after learning derivative rules.
Before this connection: Differentiation tools; Indeterminate quotient forms and one-sided limits.
Check understanding
Use the complete prompt to make student reasoning visible, then consult the separate worked reasoning and criteria. A reading completion tick records reading history.
Connect four representations of a limit →
Relate a removable-hole formula to nearby values, a graph feature, limit notation, and a verbal conclusion.
Test all three continuity conditions →
Identify the actual target value, the two-sided limit, and whether they agree.
Justify a root without locating it →
State continuity and bracket zero with endpoint values to justify existence in the open interval.
Standards and earlier reading lists
These topic placements organize reading. Edition-qualified Standards mappings and the existing complete unit task sets retain their reviewed scope.