AP Calculus AB: Limits at Infinity and Horizontal Asymptotes

Understanding how a function behaves as its inputs become extremely large or small is a cornerstone of calculus. This analysis of end behavior—what happens to $f(x)$ as $x$ approaches positive or negative infinity—allows you to sketch graphs accurately, model long-term trends, and is fundamental to later concepts like improper integrals. The primary tool for this analysis is evaluating limits at infinity, which directly reveals the presence of horizontal asymptotes.

Defining Limits at Infinity and Horizontal Asymptotes

Formally, we say the limit of $f(x)$ as $x$ approaches infinity is $L$ if the values of $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently large. We write this as ${\lim }_{x\to \infty }f(x)=L$. The same concept applies for negative infinity: ${\lim }_{x\to -\infty }f(x)=M$.

A horizontal asymptote is a horizontal line $y=L$ that the graph of the function approaches as $x$ tends to $\infty$ or $-\infty$. Crucially, a function can cross a horizontal asymptote. The asymptote describes the function's end behavior, not its behavior at finite values of $x$. For example, the function $f(x)=\frac{3x}{x^2+1}$ has a horizontal asymptote at $y=0$ because ${\lim }_{x\to \infty }\frac{3x}{x^2+1}=0$ and ${\lim }_{x\to -\infty }\frac{3x}{x^2+1}=0$, even though the graph crosses the x-axis (which is $y=0$) at $x=0$.

Evaluating Limits at Infinity for Rational Functions

The most common technique for rational functions (ratios of polynomials) is to divide the numerator and denominator by the highest power of $x$ present in the denominator. This standardizes the expression and makes the limit evident.

Example 1: Find ${\lim }_{x\to \infty }\frac{2x^3-5x}{4x^3+1}$.

1. The highest power of $x$ in the denominator is $x^3$.
2. Divide every term in the numerator and denominator by $x^3$:

$$\underset{x\to \infty }{\lim }\frac{(2x^3/x^3)-(5x/x^3)}{(4x^3/x^3)+(1/x^3)}=\underset{x\to \infty }{\lim }\frac{2-(5/x^2)}{4+(1/x^3)}$$

1. As $x\to \infty$, terms like $5/x^2$ and $1/x^3$ approach $0$.
2. Therefore, the limit simplifies to $\frac{2-0}{4+0}=\frac{1}{2}$. The horizontal asymptote is $y=\frac{1}{2}$.

The end behavior of a rational function follows a general rule based on the degrees of the numerator ($n$) and denominator ($m$):

• If $n<m$: The limit is $0$. The horizontal asymptote is $y=0$.
• If $n=m$: The limit is the ratio of the leading coefficients. The horizontal asymptote is $y=\frac{a_n}{b_m}$.
• If $n>m$: The limit is $\infty$ or $-\infty$ (there is no finite horizontal asymptote). You would then analyze the behavior more closely, often finding an oblique asymptote.

Comparing Growth Rates: The Hierarchy of Functions

A powerful, non-computational way to evaluate limits at infinity is understanding the relative growth rates of different function families as $x\to \infty$. From slowest-growing to fastest-growing, a fundamental hierarchy is:

1. Constant functions
2. Logarithmic functions (e.g., $\ln x$)
3. Polynomial functions (higher degree grows faster than lower degree)
4. Exponential functions (with base $b>1$, e.g., $2^x$, $e^x$)
5. Factorial functions ($x!$)

This hierarchy lets you quickly determine limits at infinity for expressions involving these different families. The faster-growing function will dominate the expression's behavior.

Example 2: Evaluate ${\lim }_{x\to \infty }\frac{e^x}{x^3}$. The numerator is an exponential function, and the denominator is a polynomial. Since exponential functions grow faster than any polynomial, the numerator dominates. As $x$ increases, $e^x$ becomes astronomically larger than $x^3$. Therefore, ${\lim }_{x\to \infty }\frac{e^x}{x^3}=\infty$.

Example 3: Evaluate ${\lim }_{x\to \infty }\frac{\ln (x)}{x}$. Here, the logarithmic function $\ln (x)$ is in the numerator and the polynomial $x$ (degree 1) is in the denominator. Since polynomials grow faster than logarithms, the denominator dominates, driving the value of the fraction toward zero: ${\lim }_{x\to \infty }\frac{\ln (x)}{x}=0$.

Limits at Infinity for Functions Involving Radicals

When functions involve radicals (often appearing in the context of difference of squares or related rates problems), a common strategy is to factor out the highest power of $x$ from under the radical. Remember that $\sqrt{x^2}=|x|$, which is crucial when $x\to -\infty$.

Example 4: Find ${\lim }_{x\to \infty }(\sqrt{x^2+3x}-x)$.

1. Factor $x^2$ out of the square root: $\sqrt{x^2(1+3/x)}$.
2. Since $x\to \infty$, $x$ is positive, so $\sqrt{x^2}=x$. The expression becomes:

$$\underset{x\to \infty }{\lim }\left(x\sqrt{1+\frac{3}{x}}-x\right)=\underset{x\to \infty }{\lim }x\left(\sqrt{1+\frac{3}{x}}-1\right)$$

1. At this point, you can use algebraic manipulation or L'Hôpital's Rule (a later calculus topic) to evaluate. The result is $\frac{3}{2}$.

Common Pitfalls

1. Misapplying the Growth Hierarchy with Negative Infinity: The hierarchy holds for $x\to \infty$. For $x\to -\infty$, exponential functions like $e^x$ approach $0$, and the behavior of polynomials depends on whether the highest degree term is even or odd. Always consider the sign. For ${\lim }_{x\to -\infty }\frac{x^5}{e^x}$, the polynomial $x^5\to -\infty$ while $e^x\to 0^{+}$, resulting in a limit of $-\infty$.

1. Forgetting Absolute Value with Radicals and Negative $x$: A critical error is simplifying $\sqrt{x^2}$ to $x$ when $x$ is negative. For $x\to -\infty$, $\sqrt{x^2}=|x|=-x$. If Example 4 asked for ${\lim }_{x\to -\infty }(\sqrt{x^2+3x}-x)$, you would factor and get $|x|\sqrt{1+3/x}-x$. Since $x$ is negative, $|x|=-x$, leading to a different computation and final answer.

1. Assuming "No Horizontal Asymptote" Means No Limit: A function can have different horizontal asymptotes on the left and right. You must evaluate ${\lim }_{x\to \infty }f(x)$ and ${\lim }_{x\to -\infty }f(x)$ separately. For $f(x)=\frac{2x}{\sqrt{x^2+1}}$, the limit as $x\to \infty$ is $2$, but the limit as $x\to -\infty$ is $-2$ (due to the absolute value issue in the radical). Therefore, it has two horizontal asymptotes: $y=2$ and $y=-2$.

1. Overcomplicating Simple Rational Functions: Before diving into long division, apply the degree comparison rule. For ${\lim }_{x\to \infty }\frac{3x^2-1}{5x^2+2x}$, since the degrees are equal ($n=m=2$), the limit is immediately the ratio of leading coefficients: $3/5$. This saves time on the AP exam.

Summary

• The limit at infinity, ${\lim }_{x\to \infty }f(x)=L$, describes the value a function approaches as its input grows without bound and defines the horizontal asymptote $y=L$.
• For rational functions, divide numerator and denominator by the highest power in the denominator, or use the degree rule: compare degrees to find the limit is $0$, the ratio of leading coefficients, or infinite.
• Understanding the growth rate hierarchy (logs < polynomials < exponentials) provides a quick, intuitive way to evaluate challenging limits at infinity involving different function families.
• Always handle limits as $x\to -\infty$ with care, especially for polynomials (check parity) and expressions under an even root (use $|x|$).
• A function can have zero, one, or two distinct horizontal asymptotes, depending on its left-hand and right-hand end behavior.