About Course

Dive into the heartbeat of calculus by exploring how functions behave as inputs approach critical values. In this focused mini-course, you will develop both an intuitive “feel” for limits—using real-world scenarios like velocity and population growth—and the algebraic prowess to compute them with confidence. You’ll learn to spot subtle discontinuities in piecewise definitions, harness the power of limit laws to tame indeterminate forms, and wield the Intermediate Value Theorem to guarantee solutions where none seem apparent. Finally, you’ll attain the rigor of a true mathematician by mastering the ε–δ definition, laying the groundwork for every proof you’ll encounter in AP Calculus AB and beyond.

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What Will You Learn?

  • Grasp the informal and formal meaning of limits through vivid, real-life analogies
  • Apply a complete set of algebraic techniques to evaluate limits reliably
  • Spot and classify every kind of discontinuity in both smooth and piecewise functions
  • Use the Intermediate Value Theorem to assert existence of solutions in diverse contexts
  • Master the ε–δ framework to write rigorous proofs that stand up to scrutiny

Course Content

1. Intuitive Introduction to Limits
Understand the core idea of “approach” before you ever write a formula. We’ll start by examining everyday examples—like how a car’s speed approaches a stop and how population models grow toward carrying capacity—to build an internal picture of what a limit really means. You’ll learn to read and sketch graphs to predict outcomes “from the side,” and you’ll see why one-sided limits are crucial when functions behave differently on each side of a boundary.

  • 1.1 Real-World Rates & Approaching Values
  • 1.2 Graphical Interpretation of Limits
  • 1.3 One-Sided Limits & Precise Notation

2. Algebraic Techniques for Evaluating Limits
Translate your intuition into action with a suite of algebraic tools. We’ll break down limit laws—sum, product, power, and more—and apply them step by step. You’ll tackle challenging 0/0 indeterminate forms by factoring, rationalizing, and clever substitutions, then extend your toolkit to analyze behavior “at infinity” and identify horizontal asymptotes. By the end, you’ll convert messy expressions into clean answers without ever guessing.

3. Continuity & Classifying Discontinuities
Build on your limit skills to define exactly what it means for a function to be unbroken—and why breakpoints matter. You’ll learn the formal definition of continuity at a point and across an interval, then quickly spot removable “holes,” jump discontinuities, and infinite “blow-ups.” Special attention is paid to piecewise-defined functions, where continuity can hide in plain sight. Real-world analogies—like a bridge with missing sections—will cement your understanding.

4. The Intermediate Value Theorem & ε–δ Rigour
Elevate your problem-solving by using the Intermediate Value Theorem to prove that solutions must exist—whether you’re finding roots or guaranteeing crossing points. Then, step behind the curtain into the world of mathematical rigor: the ε–δ definition of a limit. You’ll see how to frame precise “challenge-response” proofs that leave no room for ambiguity, preparing you for the proof-based questions on the AP exam and future courses.

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