Problem 37
Question
Find the horisontal asymptote, if there is one, of the graph of each rational function. $$f(x)=\frac{12 x}{3 x^{2}+1}$$
Step-by-Step Solution
Verified Answer
The horizontal asymptote of the function \(f(x)=\frac{12 x}{3 x^{2}+1}\) is \(y=0\).
1Step 1: Identify the Highest Power in Numerator and Denominator
Looking at the given function, the highest power in the numerator is 1 (from \(12x\)) and in the denominator, the highest power is 2 (from \(3x^{2}\)).
2Step 2: Compare the Highest Powers
Since the highest power in the denominator (2) is greater than the highest power in the numerator (1), this means that as \(x\) approaches infinity, the denominator grows faster than the numerator and the whole function tends towards 0.
3Step 3: State the Horizontal Asymptote
Therefore, the horizontal asymptote of the function \(f(x)=\frac{12x}{3x^{2}+1}\) is \(y=0\). This means that as \(x\) becomes very large (either positively or negatively), the function \(f(x)\) approaches 0 but never actually reaches it.
Key Concepts
Rational FunctionsAsymptotic BehaviorLimits in Calculus
Rational Functions
Rational functions, such as the function in our exercise \( f(x)=\frac{12x}{3x^{2}+1} \), are expressions that relate two polynomials through division. They take the general form \( \frac{P(x)}{Q(x)} \) where \(P(x)\) and \(Q(x)\) are polynomials, and it is required that \( Q(x) \) is not zero for the function to be defined.
In understanding these functions, it's crucial to consider the degree (or highest power) of the polynomials in the numerator and the denominator, as it informs us about the behavior of the function as \( x \) tends to very large positive or negative values. This behavior is particularly important when identifying horizontal asymptotes, which we discuss in relation to their asymptotic behavior.
In understanding these functions, it's crucial to consider the degree (or highest power) of the polynomials in the numerator and the denominator, as it informs us about the behavior of the function as \( x \) tends to very large positive or negative values. This behavior is particularly important when identifying horizontal asymptotes, which we discuss in relation to their asymptotic behavior.
Asymptotic Behavior
The concept of asymptotic behavior in mathematics describes the way a function behaves as it gets close to a certain line or point without actually reaching it. When discussing rational functions, horizontal asymptotes are a key feature in illustrating this behavior.
Horizontal asymptotes represent a horizontal line that the graph of a function approaches as \( x \) moves towards positive or negative infinity. In our example, \( f(x)=\frac{12x}{3x^{2}+1} \), we determined that the horizontal asymptote is \( y=0 \) because the degree of the polynomial in the denominator is larger than that of the numerator, causing the function values to get closer and closer to zero the further away \( x \) gets from the origin. Think of it as the function leveling out into a straight, horizontal path as we look further and further along the \( x \) axis, emphasizing its long-term behavior.
Horizontal asymptotes represent a horizontal line that the graph of a function approaches as \( x \) moves towards positive or negative infinity. In our example, \( f(x)=\frac{12x}{3x^{2}+1} \), we determined that the horizontal asymptote is \( y=0 \) because the degree of the polynomial in the denominator is larger than that of the numerator, causing the function values to get closer and closer to zero the further away \( x \) gets from the origin. Think of it as the function leveling out into a straight, horizontal path as we look further and further along the \( x \) axis, emphasizing its long-term behavior.
Limits in Calculus
The study of limits is a fundamental concept in calculus that deals with understanding the behavior of functions as they approach a specific point. In the context of this exercise, we use limits to find the horizontal asymptote of a rational function by analyzing what happens as \( x \) approaches infinity (\( \infty \) or \( -\infty \) ).
By defining the limit of \( f(x) \) as \( x \) approaches infinity, we are essentially trying to pinpoint what value \( f(x) \) is getting closer to. For \( f(x)=\frac{12x}{3x^{2}+1} \), the limit is 0 as \( x \) approaches both positive and negative infinity due to the polynomial in the denominator growing much faster than the one in the numerator, forcing the overall value of \( f(x) \) to decrease and approach zero.
Understanding the concept of limits is crucial for successfully identifying horizontal asymptotes and is a critical skill in analyzing the behavior of functions within calculus.
By defining the limit of \( f(x) \) as \( x \) approaches infinity, we are essentially trying to pinpoint what value \( f(x) \) is getting closer to. For \( f(x)=\frac{12x}{3x^{2}+1} \), the limit is 0 as \( x \) approaches both positive and negative infinity due to the polynomial in the denominator growing much faster than the one in the numerator, forcing the overall value of \( f(x) \) to decrease and approach zero.
Understanding the concept of limits is crucial for successfully identifying horizontal asymptotes and is a critical skill in analyzing the behavior of functions within calculus.
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