Limits
Understand the limits of sequences and functions, and build the foundation for calculus.
Sequences
The Index
Recursive Sequences
Increasing or Decreasing
Bounded
Decimal Expansions
Approximating a Number
Level Check
Sequences as Functions
Properties on a Graph
Starting Index
Arithmetic Sequences
Geometric Sequences
Geometric Behavior
Level Check
Hitting the Target
The Limit
Any Target Size
Looping Toward the Target
Comparing Powers
Comparing Coefficients
Limits on a Graph
Finitely Many Exceptions
Leaving the Target
The Limit Does Not Exist
Goes To Infinity
Goes To Negative Infinity
Infinity on a Graph
Comparing Geometric Sequences
Vertical Asymptotes
Piecewise Functions
One-Sided Limits
Finding One-Sided Limits
Level Review
The Limit
Approaching with Intervals
Defining Intervals
The Limit Definition
Limit vs Value
The One-Sided Definition
Building Functions
Level Review
To Infinity
To Minus Infinity
Disagreeing Sides
Oscillations
Classifying Limits
Finitely Many Points
Level Review
Sweeping to Infinity
Converging at Infinity
Infinity at Infinity
Oscillations at Infinity
Limits at Negative Infinity
Level Review
Sum Rule for Limits
(Coming soon)
Dangerous Sums
(Coming soon)
Product Rule for Limits
(Coming soon)
Dangerous Products
(Coming soon)
Power Rule for Limits
(Coming soon)
Dangerous Powers
(Coming soon)
Exponential Functions
(Coming soon)
Polynomials
(Coming soon)
Rational Functions
(Coming soon)
Simplifications
(Coming soon)
The Squeeze Theorem
(Coming soon)
Trigonometric Functions
(Coming soon)
Squeezing Sine
(Coming soon)
Squeezing Cosine
(Coming soon)
Continuity at a Point
(Coming soon)
Jump Discontinuities
(Coming soon)
Removable Discontinuities
(Coming soon)
Infinite Discontinuities
(Coming soon)
Eliminating Discontinuities
(Coming soon)
Continuous on an Interval
(Coming soon)
Continuous Everywhere
(Coming soon)
Algebra of Continuity
(Coming soon)
Discontinuities in Measurements
(Coming soon)
The Intermediate Value Theorem
(Coming soon)
Using the IVT
(Coming soon)
Proving the Solutions Exist
(Coming soon)
Root Finding
(Coming soon)
Approximating the Solution
(Coming soon)
Course description
In this course, you'll build an understanding of limits — a foundational idea in calculus. Starting with sequences on number lines and graphs, you'll develop intuition for convergence before formalizing what it means for a limit to exist. You'll then extend these ideas to functions, explore continuity, and apply the Intermediate Value Theorem.
Topics covered
- Sequences
- Limits of Sequences
- Limits of Functions
- Convergence
- Divergence
- Continuity of Functions
- Intermediate Value Theorem
- Euler's Number
Prerequisites and next steps
You should start the course with a foundation in algebra and the basics of functions, including graphing equations. When you finish, you'll be well-prepared to learn about derivatives and integrals.