Connecting Limits at Infinity and Horizontal Asymptotes - AP Calculus BC Study Guide

The Core Idea: Connecting Limits at Infinity and Horizontal Asymptotes

This topic establishes the fundamental connection between a core concept in calculus—the limit—and a key graphical feature—the asymptote. Specifically, it explores the long-term or "end" behavior of a function. As the input variable $x$ grows infinitely large in either the positive or negative direction, what value, if any, does the function's output $f(x)$ approach? This limiting value defines the function's end behavior.

A horizontal asymptote is the graphical manifestation of this end behavior. It is a horizontal line, $y=L$, that the graph of the function approaches as $x\to \infty$ or as $x\to -\infty$. Therefore, finding horizontal asymptotes is not merely a graphical exercise; it is an analytical process of evaluating limits at infinity. For certain classes of functions, such as rational functions, this process can be streamlined by comparing the leading terms of the numerator and denominator to quickly determine the end behavior.

Key Definitions

The concept of a horizontal asymptote is formally defined using the language of limits. A function can have at most two distinct horizontal asymptotes: one describing its right-end behavior and one describing its left-end behavior.

Definition of a Horizontal Asymptote

The horizontal line $y=L$ is a horizontal asymptote of the graph of a function $y=f(x)$ if at least one of the following limit statements is true:

$$\underset{x\to \infty }{\lim }f(x)=L$$

or

$$\underset{x\to -\infty }{\lim }f(x)=L$$

Here, $L$ must be a finite real number. If the limit as $x$ approaches $\infty$ or $-\infty$ is $\infty$ or $-\infty$, the function does not have a horizontal asymptote in that direction.

Understanding End Behavior

A horizontal asymptote is a precise mathematical description of a function's end behavior. It answers the question: "Where are the $y$-values of the function headed as the $x$-values get unboundedly large or small?"

• Right-End Behavior: The limit as $x\to \infty$ determines the behavior of the function on the far-right side of the coordinate plane. If ${\lim }_{x\to \infty }f(x)=L_1$, the graph of $f(x)$ will get arbitrarily close to the horizontal line $y=L_1$ as $x$ increases without bound.

• Left-End Behavior: The limit as $x\to -\infty$ determines the behavior of the function on the far-left side of the coordinate plane. If ${\lim }_{x\to -\infty }f(x)=L_2$, the graph of $f(x)$ will get arbitrarily close to the horizontal line $y=L_2$ as $x$ decreases without bound.

It is critical to note that $L_1$ and $L_2$ may be different values, resulting in two distinct horizontal asymptotes. It is also possible for a function to have a horizontal asymptote in one direction but not the other. A common misconception is that a function cannot cross its horizontal asymptote; however, a function can cross its horizontal asymptote any number of times. The asymptote only governs the behavior of the function as $x$ approaches $\pm \infty$.

Core Concepts & Rules

• Formal Definition: The line $y=L$ is a horizontal asymptote for the function $f(x)$ if the limit of $f(x)$ as $x$ approaches positive infinity is $L$, or if the limit of $f(x)$ as $x$ approaches negative infinity is $L$.

• End Behavior: Horizontal asymptotes provide a complete description of the end behavior of a function's graph. They indicate the finite value that the function's outputs approach as the inputs grow without bound.

• Rational Functions Shortcut: For a rational function $f(x)=\frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomials, the limit at infinity can be found by analyzing the ratio of the leading terms. Let the leading term of $p(x)$ be $a{x}^n$ and the leading term of $q(x)$ be $b{x}^m$.

  • If the degree of the numerator is less than the degree of the denominator ($n<m$), then ${\lim }_{x\to \pm \infty }f(x)=0$. The horizontal asymptote is $y=0$.

  • If the degree of the numerator is equal to the degree of the denominator ($n=m$), then ${\lim }_{x\to \pm \infty }f(x)=\frac{a}{b}$. The horizontal asymptote is $y=\frac{a}{b}$, the ratio of the leading coefficients.

  • If the degree of the numerator is greater than the degree of the denominator ($n>m$), then ${\lim }_{x\to \pm \infty }f(x)=\pm \infty$. There is no horizontal asymptote.

Step-by-Step Example 1: Rational Function Application

Problem: Find the horizontal asymptote(s) of the function $f(x)=\frac{10x^3-3x}{5x^3+x^2+1}$.

Step 1: Identify the function type and the goal.

The function $f(x)$ is a rational function. To find horizontal asymptotes, we must evaluate the limits as $x\to \infty$ and $x\to -\infty$.

Step 2: Evaluate the limit as $x\to \infty$ using the leading terms.

According to the rules for rational functions, we can determine the limit at infinity by examining the ratio of the leading terms of the numerator and the denominator.

• Leading term of the numerator: $10x^3$

• Leading term of the denominator: $5x^3$

The degrees are equal (both are 3). Therefore, the limit is the ratio of their coefficients.

$$\underset{x\to \infty }{\lim }f(x)=\underset{x\to \infty }{\lim }\frac{10x^3-3x}{5x^3+x^2+1}=\frac{10}{5}=2$$

Step 3: Evaluate the limit as $x\to -\infty$.

For rational functions, the limit as $x\to -\infty$ is the same as the limit as $x\to \infty$.

$$\underset{x\to -\infty }{\lim }f(x)=\underset{x\to -\infty }{\lim }\frac{10x^3-3x}{5x^3+x^2+1}=\frac{10}{5}=2$$

Step 4: State the conclusion.

Since both the limit to positive infinity and the limit to negative infinity are 2, the function has one horizontal asymptote.

Answer: The horizontal asymptote is the line $y=2$.

Step-by-Step Example 2: Exam-Style Application with Two Asymptotes

Problem: Find all horizontal asymptotes of the function $g(x)=\frac{e^x-5}{3e^x+2}$.

Step 1: Set up the limit to positive infinity.

To find the right-end behavior, we must evaluate ${\lim }_{x\to \infty }g(x)$.

$$\underset{x\to \infty }{\lim }\frac{e^x-5}{3e^x+2}$$

As $x\to \infty$, the term $e^x$ also goes to $\infty$. This gives the indeterminate form $\frac{\infty }{\infty }$.

Step 2: Evaluate the limit to positive infinity.

We can use the same technique as for rational functions: divide the numerator and denominator by the fastest-growing term, which is $e^x$.

$$\underset{x\to \infty }{\lim }\frac{\frac{e^x}{e^x}-\frac{5}{e^x}}{\frac{3e^x}{e^x}+\frac{2}{e^x}}=\underset{x\to \infty }{\lim }\frac{1-\frac{5}{e^x}}{3+\frac{2}{e^x}}$$

As $x\to \infty$, $e^x\to \infty$, so $\frac{5}{e^x}\to 0$ and $\frac{2}{e^x}\to 0$.

$$=\frac{1-0}{3+0}=\frac{1}{3}$$

This means $y=\frac{1}{3}$ is a horizontal asymptote.

Step 3: Set up the limit to negative infinity.

To find the left-end behavior, we must evaluate ${\lim }_{x\to -\infty }g(x)$.

$$\underset{x\to -\infty }{\lim }\frac{e^x-5}{3e^x+2}$$

Step 4: Evaluate the limit to negative infinity.

This limit is not indeterminate. We must recall the end behavior of the exponential function: as $x\to -\infty$, $e^x\to 0$. We can substitute this limiting value directly.

$$\underset{x\to -\infty }{\lim }\frac{e^x-5}{3e^x+2}=\frac{0-5}{3(0)+2}=-\frac{5}{2}$$

This means $y=-\frac{5}{2}$ is also a horizontal asymptote.

Step 5: State the final conclusion.

The function has different limiting values as $x\to \infty$ and $x\to -\infty$. Therefore, it has two distinct horizontal asymptotes.

Answer: The horizontal asymptotes are the lines $y=\frac{1}{3}$ and $y=-\frac{5}{2}$.

Using Your Calculator

While determining horizontal asymptotes is an analytical process based on limits, a graphing calculator is an excellent tool for verifying your answer or exploring a function's end behavior visually.

To verify the horizontal asymptotes of $g(x)=\frac{e^x-5}{3e^x+2}$ from Example 2:

1. Graphical Verification:

• Enter Y_1 = (e^{\wedge}(X) - 5) / (3e^{\wedge}(X) + 2) into the graphing editor.

• Enter $Y_2=1/3$ and $Y_3=-5/2$ to graph your calculated asymptotes.

• Use the ZOOM feature (e.g., $ZoomFit$ or $ZoomStd$) to get an initial view.

• To see the end behavior, adjust the WINDOW. Set $Xmin$ to a large negative number (e.g., $-20$) and $Xmax$ to a large positive number (e.g., $20$).

• Observe the graph. You should see the graph of $Y_1$ approaching the line $Y_3=-2.5$ on the far left and approaching the line $Y_2\approx 0.333$ on the far right.

2. Table Verification:

• Press [2nd] then [TBLSET].

• To check the limit as $x\to \infty$, set TblStart to a large number (e.g., $100$) and ΔTbl to a large step (e.g., $100$). Go to the [TABLE] and observe that the $Y_1$ values are approaching $\mathrm{0.333...}$.

• To check the limit as $x\to -\infty$, set TblStart to a large negative number (e.g., $-100$) and ΔTbl to a large negative step (e.g., $-100$). Go to the [TABLE] and observe that the $Y_1$ values are approaching $-2.5$.

AP Exam Quick Hit

Common Question Types

• Direct Calculation from a Rational Function: Find the equation of the horizontal asymptote of $f(x)=\frac{4-3x^2}{x^2+1}$. This tests the direct application of the leading term/degree comparison rule. The answer is $y=-3$.

• Calculation with Non-Rational Functions: Find all horizontal asymptotes of $g(x)=\frac{\sqrt{9x^2+1}}{2x-5}$. This requires evaluating limits to $\infty$ and $-\infty$ separately, paying close attention to the fact that $\sqrt{x^2}=|x|$. The answers are $y=3/2$ and $y=-3/2$.

• Conceptual Justification: A function $h(x)$ is continuous and ${\lim }_{x\to \infty }h(x)=-2$. Which of the following statements must be true? The correct statement would be "The graph of $y=h(x)$ has a horizontal asymptote at $y=-2$." This directly tests the definition.

Common Mistakes

• Only Checking $x\to \infty$: Many students find ${\lim }_{x\to \infty }f(x)$ and incorrectly assume the limit as $x\to -\infty$ is the same. This is only guaranteed for rational functions. Always check both directions for functions involving radicals or exponentials.

• Incorrectly Handling Radicals: A frequent algebraic error is assuming $\sqrt{x^2}=x$ when evaluating limits as $x\to -\infty$. The correct identity is $\sqrt{x^2}=|x|$, which means for $x<0$, $\sqrt{x^2}=-x$. This sign error leads to incorrect asymptotes.

• Improper Notation: Stating an asymptote as just a number (e.g., "the asymptote is 5") instead of the full equation of the line ($y=5$). An asymptote is a line and requires an equation.

• Applying Rational Function Rules to All Functions: Students may try to compare "degrees" of non-polynomial functions, like comparing $e^x$ and $x^2$. While the concept of dominant terms is valid, the specific degree rules only apply to rational functions. Other functions require direct limit evaluation.