CCSS.MATH.HSG.CO.C.9 / TEKS G.5.B.2✓ NSW Education Standards Authority (NESA) AlignedAcademic Year 2026–27
Calculus: Conceptual Limits & Continuity
Evaluate one-sided and two-sided limits analytically, numerically, and graphically, and determine continuity.
Why It Matters:The mathematical foundation of all modern engineering, signal processing, and machine learning gradients.
Curated Video Instruction
● Verified Active EmbedSTEP-BY-STEP PROOF
Worked Example
Evaluate lim_{x->3} (x^2 - 9)/(x - 3):
1
Direct substitution test
(3^2 - 9)/(3 - 3) = 0/0 (indeterminate)
Direct evaluation yields indeterminate form, requiring algebraic simplification.
2
Factor numerator as difference of squares
(x - 3)(x + 3) / (x - 3)
Factor x^2 - 9 into (x - 3)(x + 3).
3
Cancel common factor and evaluate limit
lim_{x->3} (x + 3) = 3 + 3 = 6
The limit as x approaches 3 is 6.
✓Verification Check
Verify approaching x=3 from left (2.99) and right (3.01): 2.99 + 3 = 5.99; 3.01 + 3 = 6.01.
5.99 ≈ 6
•
6.01 ≈ 6
⚠️ Common Pitfalls & Misconceptions
Assuming 0/0 means the limit does not exist
✗ Concluding undefined upon getting 0/0✓ 0/0 is an indeterminate form; simplify algebraically (factoring/rationalizing) to find the limit
0/0 means more work is needed.
INTERACTIVE DRILLS
Guided Practice
Practice Question1 of 1
Step-by-step Feedback
Evaluate lim_{x->2} (4x + 1).
MASTERY BENCHMARK
Official Assessment Quiz
Formal Assessment
Calculus: Conceptual Limits & Continuity
Passing Criteria:70% or higher
Question 1 of 1
What is lim_{x->inf} (1/x)?
Education to Career Navigation
Where This Lesson Leads
AI Researchers, Robotics Engineers, and Financial Modelers optimize algorithms using limits and derivatives.