Understanding Limits

At the end of the lesson on linear and nonlinear functions, we fixed a first point on our curve, drew a secant line between that first point and a second point, and observed what happened when we moved the second point closer to the first point along the curve. We saw the larger the interval between the two points on the x-axis, the more the steepness of the secant line diverged from the steepness of the curve. We also observed the closer the interval, the more the secant line started to match the steepness at the first point on the curve.

In this lesson, we’ll formalize the idea of slope further and learn how to calculate the slope for nonlinear equations at any given point. You will also learn about what a limit is and what it means to be either an undefined limit or a defined limit. Not only will you learn about limits, you will learn the different properties of limits and how to apply them to calculate limits for various functions. You will also get acclimated with SymPy to calculate limits using Python. SymPy is a suite of functions that lets us calculate limits.

As you work through each concept, you’ll get to apply what you’re learning by writing code right in your browser. The Python environment inside of this course includes answer checking so you can ensure that you’ve fully mastered each concept before learning the next.


  • Learn how to represent slope using limits.
  • Learn the difference between defined and undefined limits.
  • Learn how to use SymPy to compute limits.

Lesson Outline

  1. Introduction To Limits
  2. Defined vs. Undefined Limits
  3. Introduction to SymPy
  4. Limits Using SymPy
  5. Properties Of Limits I
  6. Properties Of Limits II
  7. Undefined Limit To Defined Limit
  8. Next Steps
  9. Takeaways

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