Foundations of Limits and Continuity

TL;DR

Limits describe the value a function approaches as its input gets arbitrarily close to a certain point, even if the function isn't defined there. Continuity means a function's graph has no breaks, jumps, or holes, which we can formally check using limits. Understanding these concepts is fundamental for calculus since derivatives and integrals are built upon them.

1. The Mental Model

Imagine walking along a path (your function's graph). A limit tells you where you'd be headed at a specific spot. If the path is continuous, you can walk through that spot without lifting your feet.

2. The Core Material

What is a Limit?

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A limit is about the "tendency" of a function. When we write $\lim_{x \to a} f(x) = L$, it means that as $x$ gets closer and closer to $a$ (from both sides, but not necessarily equal to $a$), the value of $f(x)$ gets closer and closer to $L$.

Think of it like this: if you plug in numbers slightly less than $a$ and numbers slightly greater than $a$ into $f(x)$, the outputs should get very close to $L$.

It doesn't matter what $f(a)$ actually is, or even if $f(a)$ exists. The limit is purely about the function's behavior near $a$.

One-Sided Limits

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Sometimes, a function behaves differently depending on which side you approach $a$ from.
* Left-hand limit: $\lim_{x \to a^-} f(x) = L_1$ means $f(x)$ approaches $L_1$ as $x$ approaches $a$ from values less than $a$.
* Right-hand limit: $\lim_{x \to a^+} f(x) = L_2$ means $f(x)$ approaches $L_2$ as $x$ approaches $a$ from values greater than $a$.

For the overall limit $\lim_{x \to a} f(x)$ to exist, the left-hand limit and the right-hand limit must be equal. If $\lim_{x \to a^-} f(x)
eq \lim_{x \to a^+} f(x)$, then $\lim_{x \to a} f(x)$ does not exist (DNE).

Properties of Limits

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Limits play nicely with arithmetic operations:
* $\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$ (Sum Rule)
* $\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)$ (Difference Rule)
* $\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)$ (Constant Multiple Rule)
* $\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$ (Product Rule)
* $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$, provided $\lim_{x \to a} g(x)
eq 0$ (Quotient Rule)

What is Continuity?

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A function $f(x)$ is continuous at a point $x=a$ if three conditions are met:
1. $f(a)$ exists: The function is defined at $a$. There's a point on the graph at $x=a$.
2. $\lim_{x \to a} f(x)$ exists: The limit as $x$ approaches $a$ from both sides is the same value. The path leads to a single destination.
3. $\lim_{x \to a} f(x) = f(a)$: The limit value is equal to the actual function value at $a$. The path's destination is exactly where the function is defined.

If any of these conditions fail, the function is discontinuous at $a$.

Types of Discontinuities

Here's how to think about the different ways a function can be discontinuous:

graph TD
    A["Function $f(x)$ discontinuous at $x=a$?"] --> B{"Is $f(a)$ defined and $\lim_{x \to a} f(x)$ exists?"}
    B -- "No, (e.g., hole or jump)" --> C{"Does $\lim_{x \to a} f(x)$ exist?"}
    C -- "No, limits don't match" --> D["Jump Discontinuity
        (e.g., piecewise function where left $
eq$ right limit)"]
    C -- "Yes, limit exists but $f(a)$ is undefined or different" --> E["Removable Discontinuity (Hole)
        (e.g., $f(x) = (x^2-1)/(x-1)$ at $x=1$)"]
    B -- "No, (e.g., vertical asymptote)" --> F{"Is $f(a)$ undefined AND $\lim_{x \to a} f(x)$ is $\pm \infty$?"}
    F -- "Yes" --> G["Infinite Discontinuity (Vertical Asymptote)
        (e.g., $f(x) = 1/x$ at $x=0$)"]
    F -- "No (usually means it's another type of breakdown)" --> H["Other undefined cases (often edge cases we simplify to earlier types)"]
    B -- "Yes, but $\lim_{x \to a} f(x) 
eq f(a)$" --> I["Removable Discontinuity (Point Discontinuity)
        (e.g., $f(x) = x^2$ for $x 
eq 0$, $f(0)=1$)"]

Techniques for Evaluating Limits

1. Direct Substitution: If $f(x)$ is a polynomial or rational function (and the denominator isn't zero) and $a$ is in its domain, just plug $a$ into $f(x)$.
   • Example: $\lim_{x \to 2} (x^2 + 3) = 2^2 + 3 = 7$.
2. Factoring and Canceling: If direct substitution gives an indeterminate form like $\frac{0}{0}$, try to factor the numerator and denominator to cancel common factors.
   • Example: $\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = \lim_{x \to 1} \frac{(x-1)(x+1)}{x-1} = \lim_{x \to 1} (x+1) = 1+1 = 2$.
3. Conjugate Multiplication: For expressions involving square roots, multiply by the conjugate (both numerator and denominator) to simplify.
   • Example: $\lim_{x \to 0} \frac{\sqrt{x+4} - 2}{x}$. Multiply by $\frac{\sqrt{x+4}+2}{\sqrt{x+4}+2}$.
     $= \lim_{x \to 0} \frac{(x+4) - 4}{x(\sqrt{x+4}+2)} = \lim_{x \to 0} \frac{x}{x(\sqrt{x+4}+2)} = \lim_{x \to 0} \frac{1}{\sqrt{x+4}+2} = \frac{1}{\sqrt{0+4}+2} = \frac{1}{2+2} = \frac{1}{4}$.
4. One-Sided Limits (for piecewise functions or functions with absolute values): Evaluate the limit from the left ($x \to a^-$) and from the right ($x \to a^+$) separately. If they're equal, the limit exists. Otherwise, it doesn't.

3. Worked Example

Let's analyze the function $f(x) = \begin{cases} x^2 & \text{if } x < 1 \ 2x - 1 & \text{if } x \ge 1 \end{cases}$ for limits and continuity at $x=1$.

1. Check $f(1)$: Using the second rule since $x \ge 1$, $f(1) = 2(1) - 1 = 1$. So, $f(1)$ exists.

2. Check $\lim_{x \to 1} f(x)$: We need to check both one-sided limits.

   • Left-hand limit: As $x \to 1^-$ (meaning $x$ is slightly less than 1), we use the top rule:
     $\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} x^2 = (1)^2 = 1$.
   • Right-hand limit: As $x \to 1^+$ (meaning $x$ is slightly greater than or equal to 1), we use the bottom rule:
     $\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (2x - 1) = 2(1) - 1 = 1$.

   Since the left-hand limit (1) equals the right-hand limit (1), the overall limit $\lim_{x \to 1} f(x)$ exists and is equal to 1.

3. Compare $\lim_{x \to 1} f(x)$ and $f(1)$: We found $\lim_{x \to 1} f(x) = 1$ and $f(1) = 1$.
   Since $\lim_{x \to 1} f(x) = f(1)$, all three conditions for continuity are met.

Therefore, $f(x)$ is continuous at $x=1$.

4. Key Takeaways

• A limit describes a function's intended value as you approach a point, not necessarily its actual value at that 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 at a point means the function is defined there, the limit exists there, and these two values are equal.
• Discontinuities can be removable (a hole), jump (different one-sided limits), or infinite (vertical asymptote).
• When evaluating limits, always try direct substitution first; if it gives $\frac{0}{0}$, try algebraic manipulation (factoring, conjugates, etc.).

Common Mistakes to Avoid:
- Assuming $f(a)$ is the