Question

Horizontal asymptotes Determine \(\lim _{x \rightarrow \infty} f(x)\) and \(\lim _{x \rightarrow-\infty} f(x)\) for the following functions. Then give the horizontal asymptotes of \(f(\text {if any})\). $$f(x)=\frac{3 x^{3}-7}{x^{4}+5 x^{2}}$$

Short answer

Question: Determine the horizontal asymptotes of the function \(f(x) = \frac{3x^3 - 7}{x^4 + 5x^2}\). Answer: The horizontal asymptote of the function is y = 0.

Step-by-step solution

Identify the highest exponent in the denominator

The highest exponent of x in the denominator is x^4.

Divide both numerator and denominator by the highest exponent of x

Divide both the numerator and the denominator by \(x^4\): $$ \frac{3x^3 - 7}{x^4 + 5x^2} \cdot \frac{1/x^4}{1/x^4} = \frac{3(x^3/x^4) - 7/x^4}{1 + 5(x^2/x^4)} $$ Simplify the function: $$ f(x) = \frac{3/x - 7/x^4}{1 + 5/x^2} $$

Find the limits as x approaches positive and negative infinity

As x approaches positive or negative infinity, the terms with x in the denominator will approach zero. $$ \lim _{x \rightarrow \infty} 3/x = 0, \lim_{x \rightarrow -\infty} 3/x = 0 \\ \lim _{x \rightarrow \infty} 7/x^4 = 0, \lim_{x \rightarrow -\infty} 7/x^4 = 0 \\ \lim _{x \rightarrow \infty} 5/x^2 = 0, \lim_{x \rightarrow -\infty} 5/x^2 = 0 $$ Therefore, $$ \lim_{x \rightarrow \infty} f(x) = \frac{0 - 0}{1 + 0} = 0 \\ \lim_{x \rightarrow -\infty} f(x) = \frac{0 - 0}{1 + 0} = 0 $$

Determine the horizontal asymptotes

The horizontal asymptotes are the lines y = k where k is the limit of the function as x approaches positive or negative infinity. As we found in step 3, the limits of the function are both 0 as x approaches positive and negative infinity. Therefore, there is a horizontal asymptote at y = 0.

Key concepts

Limits of functions

Limits of functions help us understand the behavior of a function as the variable approaches a particular point or infinity. In this exercise, we are particularly interested in what happens as the variable \(x\) tends towards both positive and negative infinity. By finding these limits, we can determine if there are asymptotes, which are lines that the graph of the function approaches but never actually touches.

Finding the limit involves simplifying the expression as \(x\) grows very large or becomes very negative. In rational functions, this typically involves identifying and comparing the degrees of the polynomials in the numerator and the denominator.

Rational functions

Rational functions are fractions where both the numerator and the denominator are polynomials. These functions can behave very differently depending on the degrees of the polynomials involved. In the provided function \( f(x) = \frac{3x^3 - 7}{x^4 + 5x^2} \), both parts of the fraction are polynomials, and the denominator has a higher degree than the numerator. This aspect is crucial for determining the behavior of the function, especially when finding horizontal asymptotes.

When simplifying these expressions, understanding the operations involved with the exponents is key. By dividing both the numerator and the denominator by the highest power, we obtain a form that makes it easier to find the limits as \(x\) approaches infinity.

Degree of polynomials

The degree of a polynomial refers to the highest power of \(x\) present in the expression. In this example, the highest degree is found in the denominator as \(x^4\). The degree of the polynomial influences how quickly the value of the function changes and is instrumental in determining the end behavior of the function.

In general, if the degree of the denominator is greater than that of the numerator, the horizontal asymptote will be the line \(y = 0\). This conclusion stems from evaluating the function as \(x\) reaches infinity, where the terms in the numerator shrink faster compared to those in the denominator.

End behavior of functions

The end behavior of functions describes how the function behaves as \(x\) moves towards positive or negative infinity. For rational functions, this behavior is tied closely to the degree of the polynomials in both the numerator and the denominator.

In this particular exercise, because the denominator \(x^4\) has a higher degree than the numerator \(3x^3\), this determines that the value of the function approaches zero when \(x\) becomes very large or very negative. Thus, the end behavior confirms the horizontal asymptote at \(y = 0\), as deduced from the limits we calculated earlier. This simple behavior can be predicted by looking at the degree of the polynomials without requiring significant calculations.