Week 4, Lecture 11: Limits at Infinity & Asymptotes
Welcome to Lecture 11: The Long Run
In Lecture 10, we used limits as a microscope to inspect exactly what happens at a specific, localized, microscopic point (like $x = 2$). But there is an entirely different way to use limits: as a telescope.
What happens to a physical system after a million years? Does an electrical circuit charge forever, or does its voltage eventually hit a hard ceiling? If you launch a spacecraft, does its velocity keep increasing infinitely, or does it flatline at a terminal speed? This is the study of end behavior.
Today, we aim our limit telescope out toward the absolute edge of the universe: $\lim_{x \to \infty}$. By the end of this lecture, you will be able to mathematically prove the existence of Horizontal Asymptotes (ceilings on long-term growth) and Vertical Asymptotes (places where the system violently explodes to infinity).
This lecture ties together our sequence work from Week 2 and our function work from Week 3. In Week 2, we tracked sequences $a_n$ as the index $n \to \infty$ jumping through whole numbers $1, 2, 3...$. Now, we track continuous functions $f(x)$ as $x \to \infty$ sweeping through every decimal on the real number line.
📌 Lecture 11 Topics
Why Infinite Growth Hits a Math Ceiling
Discover how limits act as a telescope to the edge of the universe. We discuss skydiver terminal velocity, the billionaire and 5 cents analogy for rational functions, infinite density, and the epic $1^\infty$ mathematical battle that creates Euler's number ($e$).
1. Limits at Infinity (Horizontal Asymptotes)
In all our previous limit problems, $x$ was approaching a fixed, physical coordinate like $x = 2$ or $x = 0$. But infinity ($\infty$) is not a coordinate. You cannot plot it on a graph, and you can never arrive there. Infinity is a direction. When we say "take the limit as $x \to \infty$", we are asking a question about long-term stability: if we let a system run forever—plugging in $x=10$, then $x=100$, then $x=1{,}000{,}000$—does the output $f(x)$ eventually settle down and flatten out at a specific height $L$?
Think of a skydiver jumping out of a plane. At first, gravity accelerates them rapidly. But as they fall, air resistance pushes back harder and harder. They don't accelerate to infinite speed; their velocity eventually flatlines at exactly 120 mph (their terminal velocity). No matter how many more seconds they fall ($x \to \infty$), their speed will never exceed that invisible ceiling. In mathematics, that long-term ceiling is called a Horizontal Asymptote.
(Myth-busting note: A common misconception is that a curve can never cross its asymptote. This is false! A function can cross its horizontal asymptote millions of times, like a dampening pendulum swinging back and forth across its resting point. A horizontal asymptote only guarantees that the curve will eventually get trapped arbitrarily close to that height in the extreme long run.)
Crossing the Horizontal Asymptote
Notice how the blue curve (representing a dampening system, like a pendulum) violently crosses the dashed red asymptote $y = L$ early on. But as $x \to \infty$, the oscillations die out, and the curve gets permanently trapped microscopically close to the horizon limit $L$.
Definition: Horizontal Asymptote
The line $y = L$ is a horizontal asymptote of the curve $y = f(x)$ if either:
💡 The Foundational Rule of Infinity
Before we can analyze massive, complicated polynomials, we need one atomic truth to build upon. Think about a single pizza of fixed size $C$. If you divide that pizza among $2$ people, they each get a half. Divide it among $100$ people, they get a crumb. If you divide that exact same finite pizza among an infinite number of people ($x \to \infty$), everyone gets exactly zero pizza.
This physical intuition translates into the most critical mathematical building block for evaluating end behavior. Any finite constant divided by an infinitely growing variable gets crushed to zero:
2. The "Dominant Power" Race (Rational Functions)
When dealing with rational functions (a polynomial divided by a polynomial) like $f(x) = \frac{3x^2 - 4x + 5}{7x^2 + 1}$, finding the limit at infinity is literally a tug-of-war between the numerator and the denominator.
Why do only the highest powers matter? Imagine a billionaire who has exactly $\$1{,}000{,}000{,}000$ and $5$ cents. If you ask them for their net worth, they don't mention the $5$ cents. As $x$ grows into the billions, lower-power terms like $-4x$ and $+5$ become mathematically irrelevant microscopic dust compared to the massive gravity of $x^2$. The limit is entirely decided by the leading terms.
To evaluate these limits rigorously, we use a single algebraic trick: divide every single term by the highest power of $x$ found in the denominator. This instantly crushes all the irrelevant lower-power terms into zero using our Foundational Rule.
Case 1: Bottom Heavy (Denominator Wins)
If the highest power is in the denominator, the bottom explodes faster than the top. The fraction collapses to $0$.
Result: Horizontal Asymptote at $y = 0$.
Case 2: A Dead Tie (Equal Powers)
If the highest power is identical on top and bottom, the limit settles at the ratio of their leading coefficients.
Result: Horizontal Asymptote at $y = 3/7$.
Case 3: Top Heavy (Numerator Wins)
If the highest power is in the numerator, the top overpowers the bottom. The function shoots to $\infty$ (or $-\infty$).
Result: No Asymptote (Limit DNE).
🎯 Worked Example: Rigorous Evaluation
Evaluate the limit rigorously: $$\lim_{x \to \infty} \frac{3x^2 - x - 2}{5x^2 + 4x + 1}$$
Step 1: Identify the highest power in the denominator. Here, it is $x^2$. Divide every single term by $x^2$.
Step 2: Simplify the fractions algebraically.
Step 3: Apply the Foundational Rule of Infinity. As $x \to \infty$, any term with an $x$ remaining in its denominator collapses to $0$.
3. Infinite Limits (Vertical Asymptotes)
In Lecture 10, we saw that when a fraction evaluates to $\frac{0}{0}$, we have a removable hole. But what happens if a fraction evaluates to a non-zero number divided by zero, like $\frac{5}{0}$?
Imagine taking a fixed $5$ units of mass and packing it into a space that is shrinking closer and closer to $0$ volume ($x = 0.1$, then $0.001$, then $0.000001$). The resulting density shoots toward infinity! This creates a Vertical Asymptote. The limit technically "Does Not Exist" (because infinity is not a real number), but we use the infinity symbol to describe the exact direction of the explosion.
Definition: Vertical Asymptote
The line $x = a$ is a vertical asymptote of the curve $y = f(x)$ if at least one of the following is true:
But how do we know if the function shoots up to $+\infty$ or plunges down to $-\infty$? Since the denominator is shrinking to zero, its sign (positive or negative) determines the direction of the explosion. We must test the left-hand and right-hand limits independently using a sign analysis chart.
Opposite Directions: $f(x) = \frac{1}{x}$
As $x \to 0^+$, we divide $1$ by a tiny positive number ($+\infty$). As $x \to 0^-$, we divide $1$ by a tiny negative number ($-\infty$).
Same Direction: $f(x) = \frac{1}{x^2}$
Because $x^2$ is always positive, dividing $1$ by a tiny positive or negative number both result in a massive positive output ($+\infty$).
🎯 Worked Example: Sign Analysis at a Vertical Asymptote
Evaluate the one-sided limits for $f(x) = \frac{x + 2}{x - 3}$ at $x = 3$.
1. The Left-Hand Limit ($x \to 3^-$): Imagine plugging in $x = 2.99$. The numerator $x + 2$ is roughly $5$ (Positive). The denominator $x - 3$ is $2.99 - 3 = -0.01$ (Tiny Negative). A positive constant divided by a tiny negative number shoots to negative infinity.
2. The Right-Hand Limit ($x \to 3^+$): Imagine plugging in $x = 3.01$. The numerator is roughly $5$ (Positive). The denominator is $3.01 - 3 = +0.01$ (Tiny Positive). A positive constant divided by a tiny positive number shoots to positive infinity.
4. The Ultimate Horizon: The Limit Definition of $e$
In Week 3, we teased that the most important constant in calculus is Euler's number, $e \approx 2.71828$. We can now formally construct $e$ using limits at infinity.
Imagine you have $1$ invested at a 100% annual interest rate. If the bank compounds your interest once a year, your wealth grows by $(1 + \frac{1}{1})^1 = \$2.00$. If they compound it every month, it grows by $(1 + \frac{1}{12})^{12} = \$2.61$. What if they compound it continuously, an infinite number of times per second? Does your wealth shoot to infinity?
Let's look at the mathematics. As $n \to \infty$, the fraction $\frac{1}{n}$ gets crushed to zero, meaning the base of our expression $(1 + \frac{1}{n})$ gets closer and closer to exactly $1$. However, the exponent $n$ is simultaneously growing to infinity. This creates one of the most famous battles in calculus: the $1^\infty$ Indeterminate Form.
- The base wants to shrink the output (because $1$ to any power is $1$).
- The exponent wants to explode the output (because any number slightly bigger than $1$ raised to infinity explodes to $\infty$).
Who wins? Neither. They perfectly balance each other out, locking into a horizontal asymptote. The total payout flatlines at exactly $e$. This is the absolute mathematical ceiling of continuous growth.
The Formal Definition of Euler's Number
Graphical Asymptote
Summary: The Edges of the Universe
Take a deep breath—you have officially mapped out the entire macro and micro landscape of limits. It is completely normal if balancing infinity, indeterminate forms, and sign-analysis felt overwhelming at first. You didn't just memorize formulas; you built a toolkit that lets you stare directly into the mathematical abyss and make sense of it.
How this bridges our journey: Think back to where we started. In Week 2, we looked at discrete sequences jumping across whole numbers toward infinity. In Week 3, we mapped out continuous functions. Here in Week 4, we unified those worlds. Lecture 10 gave you the microscope to inspect localized tears and holes ($0/0$), and today's Lecture 11 gave you the telescope to look outward toward long-term ceilings ($y = L$) and vertical explosions ($\pm\infty$). Every algebraic trick you learned—factoring, rationalizing, dividing by dominant powers—was just building your intuition for how systems behave when pushed to their absolute limits.
🎓 Key Takeaways from Lecture 11
- Limits at Infinity ($x \to \infty$): Tell us the long-term end behavior of the function, revealing Horizontal Asymptotes ($y = L$).
- The Dominant Power Race: For rational functions, divide every term by the highest power of $x$ in the denominator to quickly crush irrelevant terms to zero.
- Infinite Limits (The $\frac{c}{0}$ Explosion): When a non-zero number is divided by zero, the function density shoots to $\pm\infty$, creating a Vertical Asymptote.
- Continuous Compounding: Euler's number $e$ is formally defined as the horizontal asymptote of $\left(1 + \frac{1}{n}\right)^n$ as $n \to \infty$.
Where we go next: All of this machinery—limits, continuity, and approaching boundaries—was never the end goal. It was the launchpad. Next week, we take everything you've learned about limits and apply it to a single, 300-year-old question: What happens when the distance between two points on a curve shrinks to zero? That single question dissolves the tangent line problem and gives birth to Differential Calculus (derivatives, rates of change, and optimization). You are entirely ready for it.
🧠 Practice Problems
Test your mastery of horizontal asymptotes, the dominant power race, and vertical asymptotes.
Launch Practice Problems →