Introduction to Limits and Functions

TL;DR

Functions describe relationships between variables, showing how one thing changes in response to another. Limits help us understand a function's behavior as its input gets incredibly close to a certain value. Together, they're foundational tools for understanding change and optimization in economics.

1. The Mental Model

Think of a function as a rule that takes an input and gives you exactly one output. A limit is like asking, "What value does this rule seem to be heading towards as I get really, really close to a specific input, even if I can't quite get there?"

2. The Core Material

Functions: The Rule Book

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A function is a relationship where every input has one and only one output. In economics, you'll see them everywhere:
* Demand function: Quantity demanded depends on price.
* Cost function: Total cost depends on quantity produced.
* Production function: Output depends on labor and capital.

We usually write a function as $y = f(x)$, where $x$ is the input, $y$ is the output, and $f$ is the rule.

Let's say you have a simple cost function: $C(q) = 10 + 2q$, where $C$ is total cost and $q$ is the quantity produced.
* If $q=0$, then $C(0) = 10 + 2(0) = 10$. (Fixed cost is 10)
* If $q=5$, then $C(5) = 10 + 2(5) = 20$.

You can input any quantity $q$ and get a unique total cost $C$.

Limits: Approaching a Value

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A limit describes the value a function "approaches" as the input approaches some specific value. It's about what happens near a point, not necessarily at the point itself.

We write a limit as $\lim_{x \to a} f(x) = L$. This means "as $x$ gets closer and closer to $a$ (from both sides), the value of $f(x)$ gets closer and closer to $L$."

Why is this useful?
1. Undefinable points: Sometimes a function isn't defined at a specific point (e.g., division by zero), but we still want to know what value it's heading towards.
2. Continuity: Limits help us understand if a function has "breaks" or "jumps."
3. Derivatives: Limits are the foundation for calculus, which measures rates of change (like marginal cost or marginal utility).

Consider the function $f(x) = \frac{x^2 - 1}{x - 1}$.
If you try to plug in $x=1$, you get $\frac{1^2 - 1}{1 - 1} = \frac{0}{0}$, which is undefined.
However, we can factor the numerator: $f(x) = \frac{(x-1)(x+1)}{x-1}$.
For any $x \neq 1$, we can cancel $(x-1)$, leaving $f(x) = x+1$.
So, as $x$ gets very close to 1 (but isn't exactly 1), $f(x)$ gets very close to $1+1=2$.
Thus, $\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2$.

Here's how a function relates its inputs to its outputs and how limits connect to observing its behavior:

graph LR
    A["Input (x)"] --> B{"Function Rule (f)"};
    B --> C["Output (f(x) or y)"];
    D["Value 'a' (Approached Input)"] --> E{"Observe f(x) as x gets near 'a'"};
    E --> F["Limit (L) - The value f(x) approaches"];
    B -- "Analyzed by" --> E;

3. Worked Example

Let's look at a marginal revenue function. Suppose a firm's total revenue is $TR(q) = 10q - 0.5q^2$.
The marginal revenue (MR) is the change in total revenue from selling one more unit. Conceptually, $MR(q) = TR(q+1) - TR(q)$.
However, in calculus, we define marginal revenue as the derivative of the total revenue function, which is fundamentally a limit.

Let's calculate the MR at $q=5$ using the change over a tiny increment $\Delta q$.
$MR(q) = \lim_{\Delta q \to 0} \frac{TR(q + \Delta q) - TR(q)}{\Delta q}$

We'll evaluate this for our $TR(q)$ function:
$TR(q + \Delta q) = 10(q + \Delta q) - 0.5(q + \Delta q)^2$
$= 10q + 10\Delta q - 0.5(q^2 + 2q\Delta q + (\Delta q)^2)$
$= 10q + 10\Delta q - 0.5q^2 - q\Delta q - 0.5(\Delta q)^2$

Now subtract $TR(q)$:
$TR(q + \Delta q) - TR(q) = (10q + 10\Delta q - 0.5q^2 - q\Delta q - 0.5(\Delta q)^2) - (10q - 0.5q^2)$
$= 10\Delta q - q\Delta q - 0.5(\Delta q)^2$

Divide by $\Delta q$:
$\frac{TR(q + \Delta q) - TR(q)}{\Delta q} = \frac{10\Delta q - q\Delta q - 0.5(\Delta q)^2}{\Delta q}$
$= 10 - q - 0.5\Delta q$

Finally, take the limit as $\Delta q \to 0$:
$MR(q) = \lim_{\Delta q \to 0} (10 - q - 0.5\Delta q)$
$= 10 - q - 0.5(0)$
$= 10 - q$

So, the marginal revenue function is $MR(q) = 10 - q$.
If $q=5$, then $MR(5) = 10 - 5 = 5$. This means that at a production level of 5 units, selling one more unit would add approximately 5 units to total revenue. The limit allowed us to precisely define this "instantaneous" rate of change.

4. Key Takeaways

• A function is a clear rule mapping each input to exactly one output.
• Limits describe the value a function approaches as its input gets infinitely close to a specific value.
• Limits are crucial for understanding function behavior at points where the function might be undefined.
• In economics, functions model relationships like demand, cost, and utility.
• Limits are the mathematical backbone for concepts like marginal cost and marginal revenue (derivatives).
• Understanding limits helps you grasp the idea of instantaneous rates of change.

Common Mistakes to Avoid:
- Don't confuse the value of a function at a point with its limit as it approaches that point; they can be different or undefined.
- Assuming a limit exists just because you can plug in a number; always consider what happens as you get close.
- Forgetting that limits must approach the same value from both the left and right sides to exist.
- Misinterpreting a function's output as its rate of change; functions give values, derivatives (built on limits) give rates of change.

5. Now Try It

Think about a simple demand function: $P(Q) = 100 - 2Q$, where P is price and Q is quantity.
1. Calculate the price when $Q=10$.
2. Imagine a new function related to this: $f(Q) = \frac{P(Q) - P(5)}{Q - 5}$. What does this function represent (think about change in price per change in quantity)?
3. Try to find the limit of $f(Q)$ as $Q \to 5$. What does this limit tell you about the demand curve's slope at $Q=5$?

Success looks like: You should get a specific number for the price at $Q=10$. For the limit, you should find a specific numerical value that represents the instantaneous rate of change of price with respect to quantity when quantity is 5.