Introduction to Limits

TL;DR

Limits describe what value a function approaches as its input gets closer and closer to a certain number, without necessarily reaching it. They're fundamental for understanding continuity, derivatives, and integrals in calculus. You'll learn how to find these "approaching" values, even when direct calculation isn't possible.

1. The Mental Model

Imagine you're walking towards a wall. A limit is like asking how high you'll be when you're just about to touch the wall, not necessarily what happens at the wall itself. It's about predicting the trend.

2. The Core Material

Limits help us understand the behavior of functions, especially when we can't just plug in a value. Sometimes plugging in a value leads to division by zero or other undefined results.

What is a Limit?

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A limit of a function $f(x)$ as $x$ approaches a number $c$ is a specific value $L$. We write this as:

$\lim_{x \to c} f(x) = L$

This means that as $x$ gets arbitrarily close to $c$ (from both sides, less than $c$ and greater than $c$), the value of $f(x)$ gets arbitrarily close to $L$.

Why Do We Need Limits?

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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, the function still has a predictable behavior around $x=1$.

We can factor the numerator: $x^2 - 1 = (x-1)(x+1)$.
So, $f(x) = \frac{(x-1)(x+1)}{x-1}$.
For any $x \neq 1$, we can cancel out the $(x-1)$ terms, leaving $f(x) = x+1$.
As $x$ gets closer to $1$, $x+1$ gets closer to $1+1=2$. So, the limit is 2.

One-Sided Limits

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Sometimes a function approaches different values depending on which direction $x$ comes from.
- Limit from the left: $\lim_{x \to c^-} f(x)$ (as $x$ approaches $c$ from values less than $c$).
- Limit from the right: $\lim_{x \to c^+} f(x)$ (as $x$ 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.

graph TD
    A["Does `f(x)` have a limit as `x` approaches `c`?"] --> B{"Is `lim_(x->c^-) f(x)` defined?"};
    B -- "No" --> D["Limit Does Not Exist (LNE)"];
    B -- "Yes" --> C{"Is `lim_(x->c^+) f(x)` defined?"};
    C -- "No" --> D;
    C -- "Yes" --> E{"Are `lim_(x->c^-) f(x)` and `lim_(x->c^+) f(x)` equal?"};
    E -- "No" --> D;
    E -- "Yes" --> F["Limit Exists: `L = lim_(x->c^-) f(x) = lim_(x->c^+) f(x)`"];

Techniques for Finding Limits

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1. Direct Substitution: If $f(x)$ is a polynomial or a rational function (where the denominator isn't zero at $c$), just plug in $c$.
2. Factoring and Canceling: As shown in the example above, factor the numerator/denominator to cancel common terms that cause division by zero.
3. Conjugates: For expressions involving square roots, multiply by the conjugate to simplify.
4. Tables/Graphs: Look at values of $f(x)$ as $x$ gets closer to $c$ from both sides, or examine the graph.

3. Worked Example

Let's find the limit: $\lim_{x \to 3} \frac{x^2 - 2x - 3}{x - 3}$

1. Try Direct Substitution: Plugging in $x=3$ gives $\frac{3^2 - 2(3) - 3}{3 - 3} = \frac{9 - 6 - 3}{0} = \frac{0}{0}$. This is an indeterminate form, meaning we need another approach.

2. Factor the Numerator: The numerator $x^2 - 2x - 3$ can be factored into $(x - 3)(x + 1)$.

3. Rewrite the Expression:
   $\frac{x^2 - 2x - 3}{x - 3} = \frac{(x - 3)(x + 1)}{x - 3}$

4. Cancel Common Factors: For $x \neq 3$, we can cancel the $(x - 3)$ term:
   $\frac{(x - 3)(x + 1)}{x - 3} = x + 1$

5. Evaluate the Limit of the Simplified Expression: Now, as $x$ approaches $3$, the simplified function $x+1$ approaches $3+1 = 4$.

So, $\lim_{x \to 3} \frac{x^2 - 2x - 3}{x - 3} = 4$.

4. Key Takeaways

• A limit describes what value a function approaches as its input gets very close to a specific number.
• You write a limit as $\lim_{x \to c} f(x) = L$.
• The function doesn't actually have to reach the value $L$ at $x=c$ for the limit to exist.
• For a limit to exist at a point, the left-hand and right-hand limits must be equal.
• Common techniques for finding limits include direct substitution, factoring/canceling, and using conjugates.
• Limits are crucial for understanding continuity, derivatives, and integrals in calculus.

Common Mistakes to Avoid:
- Don't confuse the value of $f(c)$ with the limit $\lim_{x \to c} f(x)$; they aren't always the same.
- Don't assume a limit exists just because you can plug in a value (check for $\frac{0}{0}$ or other undefined forms).
- Forgetting to check both one-sided limits when dealing with piecewise functions or functions that might behave differently from each side.
- Incorrectly simplifying expressions, especially when canceling terms that are zero at the limit point.

5. Now Try It

Find the limit for the function $g(x) = \frac{x^2 - 4}{x + 2}$ as $x$ approaches $-2$. What value does the function approach, even though you can't directly plug in $x=-2$? Your success looks like simplifying the expression to remove the division by zero and then substituting.