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

The Core Idea: Connecting Limits at Infinity and Horizontal Asymptotes

This topic extends the concept of a limit to describe the long-term or "end" behavior of a function. Instead of examining a function's behavior as $x$ approaches a specific finite number, we investigate what happens to the function's output values ($y$-values) as the input `x$ grows infinitely large in the positive direction ($x\to \infty$) or infinitely large in the negative direction ($x\to -\infty$).

The fundamental connection is that if a function's values approach a finite number $L$ as $x$ goes to infinity or negative infinity, the graph of the function will level off and approach the horizontal line $y=L$. This line is known as a horizontal asymptote. Therefore, finding the limit of a function at infinity is the analytical method for determining its horizontal asymptotes. For rational functions, this end behavior can be quickly determined by comparing the degrees of the polynomial in the numerator and the denominator.

Key Definitions

The core of this topic rests on two interconnected definitions derived from the concept of limits at infinity.

1. Limit at Infinity:

The expression ${\lim }_{x\to \infty }f(x)=L$ means that the values of $f(x)$ can be made arbitrarily close to the finite number $L$ by taking $x$ to be sufficiently large and positive.

Similarly, ${\lim }_{x\to -\infty }f(x)=M$ means that the values of $f(x)$ can be made arbitrarily close to the finite number $M$ by taking $x$ to be sufficiently large in magnitude and negative.

2. Horizontal Asymptote:

The line $y=L$ is a horizontal asymptote of the graph of a function $y=f(x)$ if either of the following is true:

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

A function can have at most two distinct horizontal asymptotes: one for $x\to \infty$ and one for $x\to -\infty$.

Understanding Rational Functions at Infinity

For rational functions (a ratio of two polynomials), the limit as $x\to \pm \infty$ is determined by the "dominant" terms, which are the terms with the highest power of $x$ in the numerator and the denominator. This comparison of degrees leads to three distinct cases.

Let $f(x)=\frac{P(x)}{Q(x)}$, where $P(x)$ is a polynomial of degree $n$ and $Q(x)$ is a polynomial of degree $m$.

• Case 1: Degree of Numerator < Degree of Denominator ($n<m$)

  The denominator grows much faster than the numerator. As $x$ becomes very large, the fraction approaches zero.

  $$\underset{x\to \pm \infty }{\lim }f(x)=0$$

  The horizontal asymptote is $y=0$.

• Case 2: Degree of Numerator = Degree of Denominator ($n=m$)

  The numerator and denominator grow at a comparable rate. The limit is the ratio of the leading coefficients. Let $a_n$ be the leading coefficient of $P(x)$ and $b_m$ be the leading coefficient of $Q(x)$.

  $$\underset{x\to \pm \infty }{\lim }f(x)=\frac{a_n}{b_m}$$

  The horizontal asymptote is $y=\frac{a_n}{b_m}$.

• Case 3: Degree of Numerator > Degree of Denominator ($n>m$)

  The numerator grows much faster than the denominator. The function's values will increase or decrease without bound. The limit does not exist as a finite number.

  $$\underset{x\to \pm \infty }{\lim }f(x)=\infty \text{or}\underset{x\to \pm \infty }{\lim }f(x)=-\infty$$

  In this case, there is no horizontal asymptote.

Core Concepts & Rules

• Limits at infinity describe the end behavior of a function as $x$ increases or decreases without bound.

• If ${\lim }_{x\to \infty }f(x)=L$ or ${\lim }_{x\to -\infty }f(x)=L$, where $L$ is a finite number, then the line $y=L$ is a horizontal asymptote of the graph of $f(x)$.

• The limit of a rational function as $x\to \pm \infty$ depends entirely on the relationship between the degree of the numerator and the degree of the denominator.

  • If $degree(numerator)<degree(denominator)$, the limit is 0.

  • If $degree(numerator)=degree(denominator)$, the limit is the ratio of the leading coefficients.

  • If $degree(numerator)>degree(denominator)$, the limit does not exist (it is $\pm \infty$), and there is no horizontal asymptote.

Step-by-Step Example 1: Basic Rational Function

Problem: Find the limit of $f(x)=\frac{4x^3-2x+1}{5x-8x^3}$ as $x\to \infty$ and identify any horizontal asymptotes.

Step 1: Identify the function type and degrees.

The function is a rational function. It's helpful to rewrite the denominator in standard form: $f(x)=\frac{4x^3-2x+1}{-8x^3+5x}$.

• The degree of the numerator $P(x)=4x^3-2x+1$ is 3.

• The degree of the denominator $Q(x)=-8x^3+5x$ is 3.

Step 2: Compare the degrees.

The degree of the numerator (3) is equal to the degree of the denominator (3). This corresponds to Case 2.

Step 3: Find the ratio of the leading coefficients.

• The leading coefficient of the numerator is 4.

• The leading coefficient of the denominator is -8.

The ratio is $\frac{4}{-8}=-\frac{1}{2}$.

Step 4: State the limit and the horizontal asymptote.

The limit is the ratio of the leading coefficients.

$$\underset{x\to \infty }{\lim }\frac{4x^3-2x+1}{-8x^3+5x}=-\frac{1}{2}$$

Because the limit as $x\to \infty$ is $-\frac{1}{2}$, the line $y=-\frac{1}{2}$ is a horizontal asymptote.

Step-by-Step Example 2: Exam-Style Application with a Radical

Problem: Find the equation(s) of the horizontal asymptote(s) of the function $g(x)=\frac{\sqrt{4x^2+5}}{3x-1}$.

Step 1: Recognize the need to check both $x\to \infty$ and $x\to -\infty$

Because of the square root, the function's behavior may differ for large positive and large negative values of $x$. We must evaluate two separate limits. Remember that $\sqrt{x^2}=|x|$.

Step 2: Evaluate the limit as $x\to \infty$ (positive $x$).

For very large positive $x$, the dominant term inside the square root is $4x^2$, and the dominant term in the denominator is $3x$.

$$\underset{x\to \infty }{\lim }\frac{\sqrt{4x^2+5}}{3x-1}$$

As $x\to \infty$, $x$ is positive, so $\sqrt{x^2}=x$. The function behaves like:

$$\underset{x\to \infty }{\lim }\frac{\sqrt{4x^2}}{3x}=\underset{x\to \infty }{\lim }\frac{2\sqrt{x^2}}{3x}=\underset{x\to \infty }{\lim }\frac{2x}{3x}=\frac{2}{3}$$

So, $y=\frac{2}{3}$ is a horizontal asymptote.

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

For very large negative $x$, the dominant terms are the same, but we must be careful with the sign.

$$\underset{x\to -\infty }{\lim }\frac{\sqrt{4x^2+5}}{3x-1}$$

As $x\to -\infty$, $x$ is negative, so $\sqrt{x^2}=|x|=-x$. The function behaves like:

$$\underset{x\to -\infty }{\lim }\frac{\sqrt{4x^2}}{3x}=\underset{x\to -\infty }{\lim }\frac{2\sqrt{x^2}}{3x}=\underset{x\to -\infty }{\lim }\frac{2|x|}{3x}=\underset{x\to -\infty }{\lim }\frac{2(-x)}{3x}=\frac{-2x}{3x}=-\frac{2}{3}$$

So, $y=-\frac{2}{3}$ is also a horizontal asymptote.

Step 4: State the final answer.

The function $g(x)$ has two horizontal asymptotes: $y=\frac{2}{3}$ and $y=-\frac{2}{3}$.

Using Your Calculator

While finding limits at infinity is an analytical process, a graphing calculator is an excellent tool for verifying your answer or exploring a function's behavior.

1. Graphical Verification:

• Press the Y= button and enter the function, for example, Y1 = (√(4X^2+5))/(3X-1).

• Press GRAPH. You may need to adjust the WINDOW to see the end behavior. A large $Xmax$ (e.g., 100) and Xmin` (e.g., -100) can be helpful. * Visually inspect the graph. You should see the function leveling off near $y = 2/3 on the right side and $y=-2/3$ on the left side.

2. Numerical Verification using the Table:

• After entering the function in Y=, go to TBLSET (2nd + WINDOW).

• Set TblStart to a large number (e.g., 1000) and ΔTbl to a large number (e.g., 1000).

• Go to TABLE (2nd + GRAPH). Observe the Y1 values as $X$ gets larger. They should get closer and closer to your calculated limit.

• For the limit to $-\infty$, repeat the process with a large negative TblStart (e.g., -1000) and a large negative ΔTbl (e.g., -1000). The Y1 values should approach the other limit.

AP Exam Quick Hit

Common Question Types

• Finding the Limit of a Rational Function: "Evaluate ${\lim }_{x\to -\infty }\frac{2x^5-3x}{7x^5+4x^2}$." This is a direct application of comparing degrees.

• Identifying Horizontal Asymptotes: "What are the equations of the horizontal asymptotes of the graph of $f(x)=\frac{1-5x^2}{x^2+9}$?" This is the same skill, just phrased differently.

• Functions with Radicals: "Find the horizontal asymptotes of $g(x)=\frac{8x-3}{\sqrt{16x^2+x}}$." This requires checking both $x\to \infty$ and $x\to -\infty$ carefully.

Common Mistakes

• Sign Errors with Radicals: The most common mistake is forgetting that $\sqrt{x^2}=|x|$. Students often incorrectly simplify $\sqrt{x^2}$ to $x$ when evaluating the limit as $x\to -\infty$, leading to a sign error in the final answer.

• Confusing Horizontal and Vertical Asymptotes: Students may incorrectly set the denominator equal to zero to find a horizontal asymptote. Remember: limits at infinity ($x\to \pm \infty$) are for horizontal asymptotes; setting the denominator to zero (where the numerator is non-zero) is for finding vertical asymptotes.

• Misidentifying the Dominant Term: In a polynomial like $3x^2-5x^4$, the dominant term is $-5x^4$, not $3x^2$. Always look for the term with the highest power of $x$, regardless of its position.

• Arithmetic Errors: Simple mistakes in dividing the leading coefficients, such as $\frac{3}{6}=2$ instead of $\frac{1}{2}$. Double-check your arithmetic.