Limits and Continuity

TL;DR

Limits describe the behavior of a function as its input approaches a certain value, not necessarily at that value. A function is continuous if you can draw its graph without lifting your pen. Understanding limits is crucial for grasping continuity and derivatives in calculus.

1. The Mental Model

Imagine you're walking towards a point on a path. A limit is where you're headed, even if there's a puddle right at that point and you can't step directly on it. Continuity means there are no breaks or jumps in the path.

2. The Core Material

When we talk about a limit, we're asking what value a function $f(x)$ gets closer and closer to as $x$ gets closer and closer to some number, say $a$. It doesn't matter what the function actually is at $a$, just what it's approaching.

We write this as $\lim_{x \to a} f(x) = L$. This means as $x$ approaches $a$, $f(x)$ approaches $L$.

One-Sided Limits

Photo by Jan van der Wolf on Pexels

Sometimes, a function might approach different values depending on whether you're coming from the left or the right.

• Left-hand limit: $\lim_{x \to a^-} f(x) = L_1$ (as $x$ approaches $a$ from values smaller than $a$)
• Right-hand limit: $\lim_{x \to a^+} f(x) = L_2$ (as $x$ approaches $a$ from values larger 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. That is, $L_1 = L_2$.

When a Limit Doesn't Exist

Photo by Markus Winkler on Pexels

A limit might not exist if:
1. The left-hand limit is not equal to the right-hand limit.
2. The function approaches infinity or negative infinity (an unbounded oscillation).
3. The function oscillates indefinitely between two fixed values.

Continuity

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

1. $f(a)$ is defined (the function exists at that point).
2. $\lim_{x \to a} f(x)$ exists (the limit from both sides is the same).
3. $\lim_{x \to a} f(x) = f(a)$ (the limit value is equal to the function's value at that point).

Essentially, if you can draw the graph of a function without lifting your pen over an interval, it's continuous on that interval. Discontinuities are points where you'd have to lift your pen.

graph TD
    A["Function $f(x)$ is continuous at $x=a$?"] --> B{"Is $f(a)$ defined?"}
    B -- "No" --> D["Not Continuous (Removable/Hole)"]
    B -- "Yes" --> C{"Does $\lim_{x \to a} f(x)$ exist?"}
    C -- "No" --> E["Not Continuous (Jump/Oscillation)"]
    C -- "Yes" --> F{"Is $\lim_{x \to a} f(x) = f(a)$?"}
    F -- "No" --> G["Not Continuous (Removable/Hole)"]
    F -- "Yes" --> H["Function is Continuous at $x=a$"]

Types of Discontinuities

Photo by Gustavo Fring on Pexels

• Removable (Hole): The limit exists, but $f(a)$ is either undefined or $f(a) \neq \lim_{x \to a} f(x)$. You could "redefine" the function at that point to make it continuous.
• Jump: The left-hand limit and right-hand limit both exist but are not equal.
• Infinite (Vertical Asymptote): The function approaches positive or negative infinity from one or both sides.

3. Worked Example

Let's examine the function $f(x) = \begin{cases} x^2 & \text{if } x < 1 \\ 3 & \text{if } x = 1 \\ x+1 & \text{if } x > 1 \end{cases}$ at $x=1$.

We need to check the three conditions for continuity:

1. Is $f(1)$ defined?
   From the definition, $f(1) = 3$. So, yes, it's defined.

2. Does $\lim_{x \to 1} f(x)$ exist?
   We need to check the one-sided limits:

   • Left-hand limit: $\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} x^2 = (1)^2 = 1$.
   • Right-hand limit: $\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (x+1) = (1)+1 = 2$.

   Since the left-hand limit ($1$) is not equal to the right-hand limit ($2$), the overall limit $\lim_{x \to 1} f(x)$ does not exist.

Since condition 2 is not met, the function $f(x)$ is not continuous at $x=1$. This is a jump discontinuity.

4. Key Takeaways

• A limit describes what a function approaches, not necessarily what it is at 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 requires three things: the function must be defined, the limit must exist, and the limit must equal the function's value.
• You can visualize a continuous function as one you can draw without lifting your pen from the paper.
• Discontinuities can be holes (removable), jumps, or vertical asymptotes (infinite).

Common Mistakes to Avoid:
- Don't assume $f(a)$ is the limit; always calculate the limit separately.
- For piecewise functions, remember to check both one-sided limits at the boundary points.
- Confusing "limit exists" with "function is continuous"; continuity is a stricter condition.
- Forgetting that a hole in a graph means the function is not defined at that point, or the limit doesn't match the function's value.

5. Now Try It

Consider the function $g(x) = \frac{x^2 - 4}{x - 2}$.
1. Find $\lim_{x \to 2} g(x)$.
2. Is $g(x)$ continuous at $x=2$? Explain why or why not using the conditions for continuity.

What success looks like: You should be able to state the limit value (if it exists) and clearly identify which continuity condition(s) are met or not met, explaining the type of discontinuity if applicable.