Foundations of Limits and Continuity

TL;DR

Limits describe what a function's output approaches as its input gets closer and closer to a certain value, without necessarily reaching it. Continuity means a function has no breaks, jumps, or holes, allowing you to draw its graph without lifting your pen. Understanding these concepts is crucial because they form the bedrock for all of calculus, especially derivatives.

1. The Mental Model

Imagine you're walking towards a destination. A limit is like asking where you're headed as you get infinitely close to that spot, even if you can't quite step on it. Continuity means your path to that destination is smooth and unbroken.

2. The Core Material

What is a Limit?

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A limit is the value that a function "approaches" as the input (x-value) gets closer and closer to some number. It doesn't care what the function actually is at that specific point, only what it's tending towards.

Think about f(x) = (x^2 - 1) / (x - 1). If you try to plug in x = 1, you'll get 0/0, which is undefined. However, we can still ask what f(x) is getting close to as x gets close to 1.

As x gets closer to 1 from both sides, f(x) gets closer to 2. So, the limit of f(x) as x approaches 1 is 2. We write this as lim (x→1) f(x) = 2.

One-Sided Limits

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Sometimes, a function approaches different values depending on which direction you come from.
* Limit from the left: lim (x→c⁻) f(x) means x approaches c from values smaller than c.
* Limit from the right: lim (x→c⁺) f(x) means x approaches c from values larger than c.

For a general limit lim (x→c) f(x) to exist, the left-hand limit must equal the right-hand limit. If they're different, the limit doesn't exist (DNE).

What is Continuity?

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A function f(x) is continuous at a point x = c if three conditions are met:

1. f(c) is defined (there's a point there).
2. lim (x→c) f(x) exists (the function approaches a single value).
3. lim (x→c) f(x) = f(c) (the limit is the actual value of the function at that point).

If any of these conditions fail, the function is discontinuous at x = c.

Here's a diagram showing different ways continuity can break:

graph TD
    A["Is f(c) defined?"] -->|Yes| B
    A -->|No| D["Discontinuity: Hole or Asymptote"]

    B["Does lim (x->c) f(x) exist?"] -->|Yes| C
    B -->|No| E["Discontinuity: Jump or Asymptote"]

    C["Does lim (x->c) f(x) = f(c)?"] -->|Yes| F["Function is Continuous at x=c"]
    C -->|No| G["Discontinuity: Hole (removable)"]

Types of Discontinuities

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• Removable Discontinuity (Hole): Occurs when lim (x→c) f(x) exists, but f(c) is undefined or f(c) is defined but doesn't equal the limit. You could "patch" the graph with a single point. Example: (x^2 - 1)/(x - 1) at x=1.
• Jump Discontinuity: Occurs when the left-hand limit and the right-hand limit exist but are not equal. The graph "jumps." Example: Piecewise functions like f(x) = x for x < 0 and f(x) = x+1 for x >= 0 at x=0.
• Infinite Discontinuity (Vertical Asymptote): Occurs when lim (x→c) f(x) is ±∞. The function shoots up or down to infinity. Example: 1/x at x=0.

3. Worked Example

Let's examine the function g(x):
g(x) = { x + 3 if x < 2
{ 5 if x = 2
{ x^2 if x > 2

We want to check for continuity at x = 2.

1. Is g(2) defined? Yes, the middle rule says g(2) = 5.
2. Does lim (x→2) g(x) exist?
   • Left-hand limit: As x approaches 2 from the left (x < 2), we use x + 3.
     lim (x→2⁻) (x + 3) = 2 + 3 = 5
   • Right-hand limit: As x approaches 2 from the right (x > 2), we use x^2.
     lim (x→2⁺) (x^2) = 2^2 = 4
     Since 5 ≠ 4, the left-hand limit does not equal the right-hand limit. Therefore, lim (x→2) g(x) does not exist.

Because the second condition for continuity is not met, g(x) is discontinuous at x = 2. Specifically, it's a jump discontinuity.

4. Key Takeaways

• A limit describes the value a function approaches, not necessarily its actual value, as the input gets very close to a specific point.
• For a limit to exist at a point, the function must approach the same value from both the left and the right sides.
• Continuity means a function's graph has no breaks, holes, or jumps, making it "smooth" in a sense.
• A function is continuous at a point if the function is defined there, the limit exists there, and the limit equals the function's value there.
• Removable discontinuities are like holes you could "patch," while jump and infinite discontinuities are more severe breaks.

Common Mistakes to Avoid:
- Confusing the function's value f(c) with its limit lim (x→c) f(x). They are often different!
- Assuming a limit exists just because the function is defined at the point.
- Forgetting to check both one-sided limits when determining if a general limit exists.
- Not checking all three conditions when determining continuity.

5. Now Try It

Consider the function h(x) = (x^2 - 4) / (x - 2).
1. Try to find h(2). What do you get?
2. Evaluate lim (x→2⁻) h(x) by plugging in values like 1.9, 1.99.
3. Evaluate lim (x→2⁺) h(x) by plugging in values like 2.1, 2.01.
4. What is lim (x→2) h(x)?
5. Is h(x) continuous at x = 2? Why or why not?

Success looks like: You've correctly identified that h(2) is undefined, but the limit exists and equals 4. You should then be able to explain that it's discontinuous at x=2 because h(2) is undefined (a removable discontinuity).