Khan Academy Unit 2.8 Calculus- Limits and Continuity Explained
What Unit 2.8 Actually Covers
Khan Academy's Unit 2.8 is where calculus starts to make sense—or at least, where it starts to feel less like random symbols floating on a page. This unit focuses on limits and continuity, which are the foundation for everything that comes after in calculus.
If you've been struggling with this unit, you're not alone. Most students hit a wall here because the textbook explanations are written by people who already understand the material. That's not helpful when you're the one trying to learn it.
Limits: The Core Idea
A limit describes what happens to a function as the input gets closer and closer to some value—without actually reaching it.
Think of it like standing at the edge of a cliff. You can get arbitrarily close to the edge, but you're not actually standing on it. A limit asks: what would happen if you kept getting closer forever?
The Formal Definition (and Why You Can Ignore It Initially)
Most textbooks open with the epsilon-delta definition. It's important for proofs, but for now, focus on the intuition:
- Limits exist when a function approaches a specific value
- Limits can exist even when the function is undefined at that point
- Limits describe behavior, not necessarily actual values
One-Sided vs. Two-Sided Limits
You need to understand the difference:
- Left-hand limit: What the function approaches as x comes in from values less than c
- Right-hand limit: What the function approaches as x comes in from values greater than c
- Two-sided limit: Exists only when both one-sided limits are equal
For the two-sided limit to exist, both sides have to agree. If they don't, the limit does not exist (DNE).
Continuity: When Graphs Behave
A function is continuous at a point if you can draw it at that point without lifting your pen. That's the informal definition, and it works for most purposes.
More formally, a function f(x) is continuous at x = c when:
- f(c) exists
- The limit as x approaches c exists
- The limit equals f(c)
Break any of these three conditions, and you have a discontinuity.
Types of Discontinuities
- Removable: A hole in the graph. You could "fill it in" with one point to make it continuous.
- Jump: The function suddenly jumps from one value to another. Think of a step function.
- Infinite: The function goes to infinity (vertical asymptote).
Being able to identify these on a graph is a common test question. Know them cold.
Limit Types Reference Table
| Type | Notation | When It Exists |
|---|---|---|
| Finite limit | lim x→c f(x) = L | Function approaches a finite number |
| Left-hand limit | lim x→c⁻ f(x) | Approaching from below only |
| Right-hand limit | lim x→c⁺ f(x) | Approaching from above only |
| Infinite limit | lim x→c f(x) = ∞ | Function grows without bound |
| Limit at infinity | lim x→∞ f(x) | What happens as x gets huge |
Common Problem Types on Khan Academy
Unit 2.8 tests a few recurring patterns. You need to recognize them:
- Direct substitution: Plug in the value. If you get a number, that's often your answer.
- Indeterminate forms (0/0): The function is undefined at that point, but a limit might still exist. Factor, rationalize, or use algebraic manipulation.
- Graphical limits: Read the y-value as x approaches the target from both sides.
- Evaluating continuity from a graph: Check if there's a hole, jump, or asymptote at the point.
How to Work Through Unit 2.8
Here's the practical approach:
- Watch the videos once without taking notes. Get the general idea.
- Watch again, pausing to work through examples yourself before Sal does.
- Start the exercises with the easiest problems first. Build confidence.
- When you hit a wrong answer, read the explanation. Understand why you missed it.
- Focus on graph interpretation skills. Many limit problems are easier when you can visualize them.
Where Students Get Stuck
The most common mistake is trying to find limits by brute force substitution every time. That works for polynomial functions, but it fails for rational functions at holes or for piecewise functions.
Another frequent error: confusing continuity with differentiability. They're related but not the same. A function can be continuous without being differentiable (sharp corners, like |x| at x=0, are the classic example).
If you're stuck on a specific problem type, search for practice problems on that exact format. Repetition beats reading explanations every time.
What Comes Next
Once you understand limits, derivative definition follows naturally. The derivative is just a special kind of limit—the limit of the difference quotient as h approaches zero.
Master this unit. Everything in differential calculus depends on it.