Introduction to Limits and Direct Substitution

TL;DR

Limits help us understand what a function is doing as it gets super close to a specific point, even if it's undefined at that point. Direct substitution is your first, easiest way to find a limit: just plug the number in. If that works and you get a real number, that's your limit!

1. The Mental Model

Imagine you're walking along a path (your function) towards a specific spot. A limit is about where you're headed as you get incredibly close to that spot, not necessarily where you actually are if there's a hole there. Direct substitution is like confidently stepping onto that spot, assuming there's no hole.

2. The Core Material

When we talk about the limit of a function, we're asking: "As the input (let's say 'x') gets closer and closer to a certain value (let's call it 'a'), what value does the output (the function's value, or 'f(x)') get closer and closer to?"

We write this mathematically as:
$$ \lim_{x \to a} f(x) = L $$
This just means "the limit of f(x) as x approaches 'a' is 'L'".

Direct Substitution: Your First Tool

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The simplest way to evaluate a limit is through direct substitution. This means you just take the value 'a' that x is approaching and plug it directly into the function f(x).

When can you use it?
You can use direct substitution if the function f(x) is "well-behaved" at x = a. This usually means:
1. Polynomials: Functions like $x^2 + 3x - 5$.
2. Rational Functions (fractions of polynomials): As long as the denominator isn't zero when you substitute 'a'.
3. Root Functions: As long as you're not taking the square root of a negative number (or similar issues with other roots).
4. Trigonometric Functions: Like $\sin(x)$ or $\cos(x)$.
5. Exponential and Logarithmic Functions: Within their defined domains.

Basically, if plugging 'a' into f(x) gives you a single, real, defined number, then that number is your limit!

The Direct Substitution Process

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graph TD
    A["Identify the Function f(x) and the value 'a' x is approaching"] --> B["Substitute 'a' directly into f(x)"];
    B --> C{{"Is the result a defined real number?"}};
    C -- "Yes" --> D["That real number is your limit!"];
    C -- "No (e.g., division by zero, sqrt of negative)" --> E["Direct substitution failed. Try other limit techniques (e.g., factoring, L'Hopital's rule)."];

Example: Using Direct Substitution

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Let's find the limit of $f(x) = 2x + 1$ as $x$ approaches 3.
$$ \lim_{x \to 3} (2x + 1) $$
Since $2x + 1$ is a polynomial, it's well-behaved. We can just substitute $x=3$:
$2(3) + 1 = 6 + 1 = 7$.
So, $\lim_{x \to 3} (2x + 1) = 7$.

When Direct Substitution Fails

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Sometimes, when you substitute 'a', you get something like $\frac{0}{0}$, $\frac{k}{0}$ (where k is a non-zero number), or other undefined forms. This doesn't mean the limit doesn't exist, it just means direct substitution isn't enough. You'd then need to use other techniques (like factoring and simplifying, or L'Hopital's Rule, which you'll learn later). But for now, if it fails, know that direct substitution isn't the answer for that specific problem.

3. Worked Example

Let's find the limit of the function $f(x) = \frac{x^2 - 4}{x + 2}$ as $x$ approaches $-1$.

1. Identify $f(x)$ and $a$:
   $f(x) = \frac{x^2 - 4}{x + 2}$
   $a = -1$

2. Attempt Direct Substitution:
   Plug $x = -1$ into $f(x)$:
   $f(-1) = \frac{(-1)^2 - 4}{(-1) + 2}$
   $f(-1) = \frac{1 - 4}{1}$
   $f(-1) = \frac{-3}{1}$
   $f(-1) = -3$

3. Evaluate the result:
   The result, $-3$, is a well-defined real number. Therefore, direct substitution worked!

4. Conclusion:
   $$ \lim_{x \to -1} \left( \frac{x^2 - 4}{x + 2} \right) = -3 $$

4. Key Takeaways

• A limit describes where a function's output is heading as its input approaches a certain value.
• Direct substitution is the simplest method to find a limit.
• To use direct substitution, just plug the 'approaching' value into the function.
• If direct substitution yields a defined real number, that number is your limit.
• This method works well for most "nice" functions like polynomials, and rational functions where the denominator isn't zero.
• If direct substitution results in an undefined form (like division by zero), it means you need to use other, more advanced limit techniques.

5. Now Try It

Find the limit of the function $g(x) = \sqrt{x+7}$ as $x$ approaches $2$. Take about 15 minutes to solve it. Success looks like you correctly applying direct substitution and identifying the numerical limit, noting why direct substitution is appropriate here.