Infinite Limits at Infinity
Textbook Reference
| Primary source | OpenStax Calculus Volume 1, Section 4.6: “Limits at Infinity and Asymptotes” |
| Direct link | https://openstax.org/books/calculus-volume-1/pages/4-6-limits-at-infinity-and-asymptotes |
| Textbook used in class | Stewart, Calculus, Section 3.4: “Limits at Infinity; Horizontal Asymptotes” |
Opening Scenario
Start with a single cell and double the population each hour. After $n$ hours you have $2^n$ cells. After $1$ hour: $2$. After $10$: $1{,}024$. After $50$: more than $10^{15}$. The number grows without any upper bound -- it is not approaching a specific value but shooting upward forever.
This is what it means for a limit to be $+\infty$ as the input grows: the output is eventually larger than any number you name, and it stays larger than that number after a point.
Quick Reference
Informal definitions.
- $\displaystyle\lim_{x \to \infty} f(x) = +\infty$: for any large number $M$, the outputs $f(x) > M$ eventually (for all large enough $x$).
- $\displaystyle\lim_{x \to \infty} f(x) = -\infty$: for any large negative number $N$, the outputs $f(x) < N$ eventually.
These “limits” are not real numbers. They describe unbounded behavior, not a value approached.
End behavior of polynomials. For $f(x) = a_n x^n + \cdots + a_0$ with $a_n \neq 0$:
| $n$ even | $a_n > 0$ | Both ends $\to +\infty$ |
|---|---|---|
| $n$ even | $a_n < 0$ | Both ends $\to -\infty$ |
| $n$ odd | $a_n > 0$ | Right end $\to +\infty$, left end $\to -\infty$ |
| $n$ odd | $a_n < 0$ | Right end $\to -\infty$, left end $\to +\infty$ |
Key Concepts
1. End Behavior of Polynomials
For large $|x|$, a polynomial is dominated by its leading term $a_n x^n$. All other terms become negligible by comparison. So the end behavior of $f(x) = a_n x^n + \cdots + a_0$ is the same as the end behavior of $a_n x^n$.
Predict first. Before computing, use the two features of the leading term:
- Degree: even or odd determines whether the two ends go the same or opposite directions.
- Leading coefficient: positive or negative determines whether the right end goes up or down.
Example 1. Describe the end behavior of $f(x) = -2x^5 + 3x^3 - x$.
Leading term: $-2x^5$. Degree $5$ is odd; leading coefficient $-2$ is negative.
Odd degree means the two ends go in opposite directions. Negative leading coefficient means the right end goes down.
$\lim_{x \to +\infty} f(x) = -\infty$ and $\lim_{x \to -\infty} f(x) = +\infty$.
Verify one direction: As $x \to +\infty$, $-2x^5 \to -\infty$. The other terms are negligible for large $x$. Confirmed.
Boxed answer: $\lim_{x \to +\infty} f(x) = -\infty$; $\lim_{x \to -\infty} f(x) = +\infty$.
2. Growth Rate Hierarchy
Some functions grow faster than others. Knowing the hierarchy lets you read off limits without computation.
Hierarchy (from slowest to fastest growth): $$\ln x \ll x^a \text{ (any } a > 0\text{)} \ll b^x \text{ (any } b > 1\text{)} \ll x^x$$
More specifically, for any positive power $a$ and any base $b > 1$:
- $\displaystyle\lim_{x \to \infty} \frac{\ln x}{x^a} = 0$ (logarithm is overwhelmed by any positive power of $x$)
- $\displaystyle\lim_{x \to \infty} \frac{x^a}{b^x} = 0$ (any power of $x$ is overwhelmed by an exponential)
Example 2. Without any computation, predict and then state: $$\lim_{x \to \infty} \frac{e^x}{x^{100}}.$$
Prediction. $e^x$ is exponential; $x^{100}$ is a power. Exponential beats any power. The fraction should grow without bound.
Conclusion: $+\infty$.
Example 3. $\displaystyle\lim_{x \to \infty} \frac{\ln x}{\sqrt{x}}$.
$\sqrt{x} = x^{1/2}$ is a positive power. Logarithm loses to any positive power.
Conclusion: $0$.
3. Two Infinite Limits That Differ
Example 4. Describe the end behavior of $f(x) = x^3 - 4x$.
Leading term: $x^3$. Odd degree, positive coefficient.
$\lim_{x \to +\infty} f(x) = +\infty$ and $\lim_{x \to -\infty} f(x) = -\infty$.
Check by intuition: for large positive $x$, $x^3$ dominates $4x$ and is positive. For large negative $x$, $x^3$ is large and negative.
4. Arithmetic with Infinity
A few rules for combining infinite limits (used when simplifying):
- $L + \infty = +\infty$ for any finite $L$.
- $L \cdot \infty = +\infty$ if $L > 0$, and $-\infty$ if $L < 0$.
- $\infty + \infty = +\infty$.
- Indeterminate forms: $\infty - \infty$, $0 \cdot \infty$, and $\infty / \infty$ cannot be resolved by these rules alone. An indeterminate form requires more work (usually algebra or L’Hopital’s Rule).
“$\infty - \infty = 0$.” This is an indeterminate form. The two quantities going to infinity might do so at different rates. For example: $x^2 - x \to +\infty$ as $x \to \infty$ (the $x^2$ dominates), and $x - x^2 \to -\infty$. Neither is zero.
Common Errors Summary
| Error | Example | Correction |
|---|---|---|
| Treating $\infty$ as a number | Writing $\lim f(x) = \infty$, then computing $\infty - \infty = 0$ | Infinity is not a number; $\infty - \infty$ is indeterminate |
| Wrong sign for negative leading coefficient | Saying right end $\to +\infty$ for $-3x^4$ | $-3x^4 \to -\infty$ as $x \to +\infty$; the negative coefficient flips the direction |
| Ignoring the degree (odd vs even) | Assuming both ends go the same way for all polynomials | Even degree: same direction; odd degree: opposite directions |
Leveled Practice
Level 1 -- Direct Application
Problem 1. State the end behavior (limits as $x \to \pm\infty$) of $f(x) = 4x^6 - x^4 + 2x$.
Show answer
Leading term: $4x^6$. Even degree, positive coefficient: both ends go to $+\infty$.
$\lim_{x \to +\infty} f(x) = +\infty$ and $\lim_{x \to -\infty} f(x) = +\infty$.
Problem 2. State the end behavior of $g(x) = -x^3 + 5x^2$.
Show answer
Leading term: $-x^3$. Odd degree, negative coefficient: right end goes down, left end goes up.
$\lim_{x \to +\infty} g(x) = -\infty$ and $\lim_{x \to -\infty} g(x) = +\infty$.
Level 2 -- Multiple Steps
Problem 3. Arrange in order of growth rate from slowest to fastest: $e^x$, $x^{10}$, $\ln x$, $x^{1/3}$.
Show answer
Hierarchy: logarithm < power < exponential. Among powers, smaller exponent grows slower.
$\ln x \ll x^{1/3} \ll x^{10} \ll e^x$.
Boxed answer: $\ln x$, then $x^{1/3}$, then $x^{10}$, then $e^x$.
Problem 4. Evaluate $\displaystyle\lim_{x \to \infty} (x^2 - x)$ without stating “infinity minus infinity is indeterminate.” Use algebra to find the actual behavior.
Show answer
Factor $x$ from both terms: $x^2 - x = x(x - 1)$.
As $x \to \infty$: both $x \to \infty$ and $x - 1 \to \infty$, so the product $x(x-1) \to +\infty$.
Boxed answer: $\displaystyle\lim_{x \to \infty} (x^2 - x) = +\infty$.
The key is to factor rather than treat the two terms separately. The form $\infty - \infty$ is indeterminate, but the algebraic form $x(x-1)$ makes the behavior clear.
Mastery Checklist
Mental Model
An infinite limit at infinity is not a number -- it is a direction report: “the outputs are heading upward (or downward) without any ceiling.” The value is never reached; what the limit records is that no finite number can serve as an upper (or lower) bound for the outputs eventually.
The growth hierarchy answers one of the most useful qualitative questions in applied math: which type of function grows fastest? Exponentials always eventually outpace any polynomial; polynomials always eventually outpace any logarithm. These are not approximations -- they are precise limit statements.
Back to Applications of Differentiation | Next: Limits at Infinity for Rational Functions