Introduction to Limits and Infinite Behavior

TL;DR

Limits at infinity help us understand how a function behaves as its input gets extremely large (positive or negative). They tell us if the function's output settles down to a specific value, grows without bound, or shrinks without bound. This concept is crucial for understanding a function's long-term trend and identifying horizontal asymptotes.

1. The Mental Model

Imagine you're driving on a very long, straight highway. Limits at infinity are like asking what your altitude or speed is as you drive forever. Does it level off? Does it keep climbing? Does it drop into a valley?

2. The Core Material

When we talk about "limits at infinity," we're really asking: what happens to the function $f(x)$ as $x$ gets arbitrarily large, either positively ($x \to \infty$) or negatively ($x \to -\infty$)?

What does "$x \to \infty$" mean?

Photo by BM Amaro on Pexels

It doesn't mean $x$ becomes infinity (infinity isn't a number you can reach). It means $x$ is growing endlessly, without any upper bound. Think of numbers like 10, 1000, 1,000,000, and so on.

What does "$x \to -\infty$" mean?

Photo by Shubham Dhage on Pexels

Similarly, this means $x$ is getting endlessly smaller, without any lower bound. Think of numbers like -10, -1000, -1,000,000, and so on.

What are the possible outcomes?

Photo by Markus Winkler on Pexels

When you evaluate a limit as $x \to \infty$ or $x \to -\infty$, there are generally three things that can happen to the function's output, $f(x)$:

graph TD
    A["As x approaches infinity"] --> B{What does f(x) do?};
    B --> C["f(x) approaches a specific number L"]
    C -- "Example: f(x) = 1/x" --> D["Horizontal Asymptote at y = L"];
    B --> E["f(x) grows without bound"]
    E -- "Example: f(x) = x^2" --> F["f(x) --> Infinity"];
    B --> G["f(x) shrinks without bound"]
    G -- "Example: f(x) = -x" --> H["f(x) --> Negative Infinity"];

• It approaches a specific number L: This means the function's graph flattens out and gets closer and closer to the horizontal line $y=L$. This line is called a horizontal asymptote.
  • Example: For $f(x) = 1/x$, as $x \to \infty$, $f(x) \to 0$. As $x \to -\infty$, $f(x) \to 0$. So, $y=0$ is a horizontal asymptote.
• It grows without bound: This means $f(x)$ gets larger and larger indefinitely. We write this as $\lim_{x \to \infty} f(x) = \infty$.
  • Example: For $f(x) = x^2$, as $x \to \infty$, $f(x) \to \infty$.
• It shrinks without bound: This means $f(x)$ gets smaller and smaller indefinitely (more and more negative). We write this as $\lim_{x \to \infty} f(x) = -\infty$.
  • Example: For $f(x) = -x$, as $x \to \infty$, $f(x) \to -\infty$.

Key Idea: Dominant Terms

Photo by Tobias Dziuba on Pexels

For rational functions (polynomial divided by polynomial), the behavior at infinity is often determined by the highest power terms in the numerator and denominator. We'll explore this more later, but the simple idea is that for very large $x$, terms like $x^2$ totally overshadow terms like $x$ or constants.

How to write it

We use the notation: $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$.

3. Worked Example

Let's consider the function $f(x) = \frac{3x - 1}{x + 2}$. We want to find $\lim_{x \to \infty} f(x)$.

1. Initial thought: If we just plug in "infinity," we get $\frac{3(\infty) - 1}{\infty + 2}$, which looks like $\frac{\infty}{\infty}$. This is an indeterminate form, meaning it doesn't immediately tell us the answer.

2. Divide by the highest power of x in the denominator: The highest power of $x$ in the denominator is $x^1$. So, we divide every term in the numerator and denominator by $x$:
   $f(x) = \frac{\frac{3x}{x} - \frac{1}{x}}{\frac{x}{x} + \frac{2}{x}}$

3. Simplify:
   $f(x) = \frac{3 - \frac{1}{x}}{1 + \frac{2}{x}}$

4. Evaluate the limit: Now, as $x \to \infty$:

   • $\frac{1}{x} \to 0$ (because 1 divided by a huge number is almost 0)
   • $\frac{2}{x} \to 0$ (for the same reason)

   So, the limit becomes:
   $\lim_{x \to \infty} \frac{3 - \frac{1}{x}}{1 + \frac{2}{x}} = \frac{3 - 0}{1 + 0} = \frac{3}{1} = 3$

Therefore, $\lim_{x \to \infty} \frac{3x - 1}{x + 2} = 3$. This means that as $x$ gets extremely large, the function's output gets closer and closer to 3. The horizontal asymptote is $y=3$.

4. Key Takeaways

• Limits at infinity describe a function's behavior as $x$ becomes extremely large (positive or negative).
• They don't involve $x$ reaching infinity, but rather $x$ growing indefinitely.
• Possible outcomes are approaching a specific number (horizontal asymptote), growing infinitely large, or shrinking infinitely small.
• Horizontal asymptotes occur when a function approaches a constant value $L$ as $x \to \pm \infty$.
• For rational functions, the ratio of the highest power terms often dictates the limit at infinity.
• Dividing by the highest power of $x$ in the denominator is a common technique to evaluate these limits.

Common mistakes to avoid:
* Treating infinity like a regular number you can substitute into an expression.
* Confusing limits at infinity with vertical asymptotes (which are about $x$ approaching a finite number where $f(x) \to \pm \infty$).
* Assuming all functions have a horizontal asymptote; some grow or shrink indefinitely.
* Forgetting to consider $x \to -\infty$ as well, as sometimes the behavior is different.

5. Now Try It

Consider the function $g(x) = \frac{5x^2 + 2x}{x^2 - 3}$. Calculate $\lim_{x \to \infty} g(x)$. Explain your steps and state whether there's a horizontal asymptote and, if so, its equation. Success looks like correctly applying the division technique and identifying the limit and asymptote.