Calculus: Limits, Continuity and Limit Laws

TL;DR

Limits help us understand a function's behavior as its input approaches a certain value, even if the function isn't defined there. Continuity means a function has no breaks or jumps, which we can check using limits. Limit laws are rules that simplify how we calculate limits of combined functions.

1. The Mental Model

Imagine you're driving towards a destination on a road. A limit describes where you're headed as you get closer and closer to that destination, without necessarily having to arrive there. Continuity is like a smooth, unbroken road without any potholes or missing sections.

2. The Core Material

When we talk about a limit, written as $\lim_{x \to c} f(x) = L$, we're asking: "What value 'L' does $f(x)$ get closer and closer to as 'x' gets closer and closer to 'c'?" It doesn't matter what $f(c)$ actually is, or even if $f(c)$ exists. We're only interested in the values around $c$.

One-Sided Limits

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Sometimes, a function behaves differently depending on which direction you approach 'c' from.
- Limit from the left: $\lim_{x \to c^-} f(x)$ means 'x' approaches 'c' from values less than 'c'.
- Limit from the right: $\lim_{x \to c^+} f(x)$ means 'x' approaches 'c' from values greater than 'c'.

For the overall limit $\lim_{x \to c} f(x)$ to exist, the left-hand limit and the right-hand limit must be equal.

Continuity

A function $f(x)$ is continuous at a point $x = c$ if three conditions are met:
1. $f(c)$ is defined (the function exists at that point).
2. $\lim_{x \to c} f(x)$ exists (the limit exists at that point).
3. $\lim_{x \to c} f(x) = f(c)$ (the limit value is equal to the function's value at that point).

Think of it as being able to draw the function's graph through $x=c$ without lifting your pen.

Here's how these concepts relate:

graph TD
    A["Consider point 'c'"] --> B{"Is f(c) defined?"}
    B -- No --> C["f(x) is not continuous at 'c'"]
    B -- Yes --> D{"Does lim(x->c) f(x) exist?"}
    D -- No --> C
    D -- Yes --> E{"Is lim(x->c) f(x) = f(c)?"}
    E -- No --> C
    E -- Yes --> F["f(x) is continuous at 'c'"]

Limit Laws

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These are rules that make finding limits of complex functions easier by breaking them down into simpler parts. Suppose $\lim_{x \to c} f(x) = L$ and $\lim_{x \to c} g(x) = M$.
1. Sum/Difference Law: $\lim_{x \to c} [f(x) \pm g(x)] = L \pm M$
2. Constant Multiple Law: $\lim_{x \to c} [k \cdot f(x)] = k \cdot L$ (where k is a constant)
3. Product Law: $\lim_{x \to c} [f(x) \cdot g(x)] = L \cdot M$
4. Quotient Law: $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{L}{M}$, provided $M \neq 0$.
5. Power Law: $\lim_{x \to c} [f(x)]^n = L^n$ (for any real number n)
6. Root Law: $\lim_{x \to c} \sqrt[n]{f(x)} = \sqrt[n]{L}$ (if $n$ is even, assume $L \ge 0$)

These laws mean you can often plug 'c' directly into a polynomial or rational function (where the denominator isn't zero at 'c') to find the limit. This is called direct substitution.

3. Worked Example

Let's evaluate $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$ and determine if $f(x) = \frac{x^2 - 4}{x - 2}$ is continuous at $x=2$.

First, try direct substitution: $\frac{2^2 - 4}{2 - 2} = \frac{0}{0}$. This is an indeterminate form, which means the limit might exist, but we need to do more work.

Let's simplify the function for $x \neq 2$:
$f(x) = \frac{(x-2)(x+2)}{x-2} = x+2$ (for $x \neq 2$).

Now, find the limit of the simplified function:
$\lim_{x \to 2} (x+2)$
Using direct substitution (which is allowed for polynomials):
$= 2+2 = 4$.

So, $\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = 4$.

Now, let's check for continuity at $x=2$:
1. Is $f(2)$ defined? $f(2) = \frac{2^2 - 4}{2 - 2} = \frac{0}{0}$. No, it's undefined.
2. Does $\lim_{x \to 2} f(x)$ exist? Yes, we found it's 4.
3. Is $\lim_{x \to 2} f(x) = f(2)$? No, because $f(2)$ is undefined.

Since $f(2)$ is not defined, the function $f(x) = \frac{x^2 - 4}{x - 2}$ is not continuous at $x=2$. It has a "hole" at $x=2$.

4. Key Takeaways

• A limit describes a function's behavior near a point, not necessarily at the point itself.
• For a limit to exist, the left-hand limit and the right-hand limit must be equal.
• Continuity at a point means the function is defined there, the limit exists there, and the limit equals the function's value.
• Limit laws allow you to break down complex limits into simpler calculations.
• Direct substitution often works for polynomials and rational functions unless it leads to an indeterminate form like $\frac{0}{0}$.
• If direct substitution results in $\frac{0}{0}$, try algebraic manipulation (like factoring) to simplify the expression before re-evaluating the limit.

Common Mistakes to Avoid:
- Confusing $f(c)$ with $\lim_{x \to c} f(x)$: They are often different, and one can exist without the other.
- Assuming a function is continuous just because its limit exists: Remember all three continuity conditions must be met.
- Ignoring one-sided limits: For piecewise functions, you must check both sides for the overall limit to exist.
- Stopping at an indeterminate form: $\frac{0}{0}$ means "do more work," not "the limit doesn't exist."

5. Now Try It

Consider the function $g(x) = \begin{cases} x^2+1 & \text{if } x < 1 \\ 3 & \text{if } x = 1 \\ 2x & \text{if } x > 1 \end{cases}$.
1. Calculate $\lim_{x \to 1^-} g(x)$.
2. Calculate $\lim_{x \to 1^+} g(x)$.
3. Does $\lim_{x \to 1} g(x)$ exist? If so, what is it?
4. Is $g(x)$ continuous at $x=1$? Explain your reasoning based on the three conditions for continuity.

Success looks like clearly showing your steps for each calculation and precisely explaining whether the function is continuous at $x=1$ based on the definitions you've learned.