Problem 17 - Thomas Calculus

Question

In Exercises \(13-22,\) find the limit of each rational function (a) as \(x \rightarrow \infty\) and \((b)\) as \(x \rightarrow-\infty\) . $$h(x)=\frac{7 x^{3}}{x^{3}-3 x^{2}+6 x}$$

Step-by-Step Solution

Answer

Both limits are 7.

Step 1: Identify the degrees of the polynomials

The given rational function is \( h(x) = \frac{7x^3}{x^3 - 3x^2 + 6x} \). Identify the degree of the polynomial in the numerator, which is 3, and the degree of the polynomial in the denominator, which is also 3.

Step 2: Determine the leading coefficients

Since the degrees of the numerator and denominator are the same, focus on their leading coefficients. The leading coefficient of the numerator is 7, and for the denominator, it is 1.

Step 3: Find the limit as \(x \rightarrow \infty\)

As \(x\) approaches infinity, the behavior of the rational function is dominated by the leading terms. Calculate the limit: \[ \lim_{x \to \infty} h(x) = \lim_{x \to \infty} \frac{7x^3}{x^3 - 3x^2 + 6x} = \frac{7}{1} = 7. \]

Step 4: Find the limit as \(x \rightarrow -\infty\)

Similarly, as \(x\) approaches negative infinity, the behavior is still governed by the leading terms. Thus, calculate the limit: \[ \lim_{x \to -\infty} h(x) = \lim_{x \to -\infty} \frac{7x^3}{x^3 - 3x^2 + 6x} = \frac{7}{1} = 7. \]

Key Concepts

Polynomial Degree

When we talk about the degree of a polynomial, we are referring to the highest power of the variable in the polynomial expression. For example, in a polynomial like \(7x^3 - 3x^2 + 6x\), the degree is 3 because the highest exponent of \(x\) is 3. In the context of rational functions, which are ratios of two polynomials, the polynomial degree of both the numerator and the denominator plays a crucial role in determining the function's behavior at infinity. It tells us which terms dominate and thus heavily influence the value of the function when \(x\) becomes very large (positively or negatively). In our example, both the numerator \(7x^3\) and the denominator \(x^3 - 3x^2 + 6x\) have a degree of 3.

Leading Coefficients

Leading coefficients are the coefficients of the highest degree terms in polynomials. These coefficients are particularly important in rational functions because they help determine the end behavior of the function. If the degree of the numerator and the degree of the denominator are the same, the leading coefficients can be used to find limits as \(x\) approaches infinity or negative infinity. For the rational function \(h(x) = \frac{7x^3}{x^3 - 3x^2 + 6x}\), the leading coefficient of the numerator is 7, and for the denominator, it is 1. The ratio of these leading coefficients gives us the limit of the function as \(x\) approaches both infinity and negative infinity. Simply put, as \(x\) grows large, the smaller terms become negligible, and the function behaves like \(\frac{7x^3}{x^3}\), simplifying to \(\frac{7}{1} = 7\).

Infinite Limits

Infinite limits describe what happens to the function's values as \(x\) goes to infinity or negative infinity. Specifically, for rational functions, we look at the relationship between the degrees of the polynomials and their leading coefficients to determine these limits. If the degree of the numerator is greater than the degree of the denominator, the limit is infinite. If the degree of the numerator is less than that of the denominator, the limit is zero. However, if both degrees are equal, like in the function \(h(x) = \frac{7x^3}{x^3 - 3x^2 + 6x}\), the limit is the ratio of the leading coefficients, which is 7. This means that no matter how large or how small \(x\) becomes, the function will tend towards 7, establishing a horizontal asymptote at \(y = 7\). This concept helps us understand the ultimate behavior of the function and predict its behavior over very large values of \(x\).