Problem 13 - Thomas Calculus

Question

In Exercises \(13-22,\) find the limit of each rational function (a) as \(x \rightarrow \infty\) and \((b)\) as \(x \rightarrow-\infty\) . $$f(x)=\frac{2 x+3}{5 x+7}$$

Step-by-Step Solution

Answer

The limits are \( \frac{2}{5} \) as \( x \rightarrow \infty \) and \( x \rightarrow -\infty \).

Step 1: Understand the Function

We are given the rational function \( f(x) = \frac{2x + 3}{5x + 7} \). Our task is to find the limit of this function as \( x \rightarrow \infty \) and \( x \rightarrow -\infty \). A rational function is in the form \( \frac{P(x)}{Q(x)} \) where both \( P(x) \) and \( Q(x) \) are polynomials. In this case, both the numerator \( 2x + 3 \) and the denominator \( 5x + 7 \) are linear polynomials.

Step 2: Find the Leading Term

To find the limit, focus on the leading term (highest degree term) of the numerator and denominator since these terms dominate the behavior of the function as \( x \rightarrow \infty \) or \( x \rightarrow -\infty \). The leading term in the numerator is \( 2x \), and in the denominator, it is \( 5x \). Therefore, as \( x \) becomes very large or very small, the function behaves like \( \frac{2x}{5x} = \frac{2}{5} \).

Step 3: Calculate \( \lim_{x \rightarrow \infty} f(x) \)

As \( x \rightarrow \infty \), the lower degree terms (\(+3\) in the numerator and \(+7\) in the denominator) become less significant. The function simplifies to \( \frac{2x}{5x} = \frac{2}{5} \). Thus, \( \lim_{x \rightarrow \infty} \frac{2x + 3}{5x + 7} = \frac{2}{5} \).

Step 4: Calculate \( \lim_{x \rightarrow -\infty} f(x) \)

Similarly, as \( x \rightarrow -\infty \), the lower degree terms \(+3\) and \(+7\) become negligible. The function still simplifies to \( \frac{2x}{5x} = \frac{2}{5} \). Therefore, \( \lim_{x \rightarrow -\infty} \frac{2x + 3}{5x + 7} = \frac{2}{5} \).

Step 5: Conclude the Limits

Both limits as \( x \rightarrow \infty \) and \( x \rightarrow -\infty \) are \( \frac{2}{5} \). This shows that the function approaches the horizontal asymptote \( y = \frac{2}{5} \).

Key Concepts

Rational Functions

Rational functions are expressions of the form \( \frac{P(x)}{Q(x)} \), where \(P(x)\) and \(Q(x)\) are polynomial functions. These functions can exhibit various interesting behaviors as their input \(x\) approaches infinity or negative infinity. Each polynomial, whether in the numerator or the denominator, has a degree, defined as the highest power of \(x\) present in the expression.
Here, you'll often encounter terms like linear, quadratic, or cubic depending on the degree:

• Linear: First-degree, such as \(2x + 3\).
• Quadratic: Second-degree, like \(x^2 + 3x + 4\).
• Cubic: Third-degree, for instance \(x^3 + x^2 + 1\).

As we observe what happens to rational functions as \(x\) approaches very large or very small values, it's the terms with the highest degree in the numerator and denominator that dominate. This is due to them growing faster than the lower-degree terms, significantly affecting the function's behavior at its extremities.

Horizontal Asymptotes

Horizontal asymptotes are lines that a function approaches as \(x\) becomes extremely large or small. For a rational function \( \frac{P(x)}{Q(x)} \), determining the horizontal asymptote involves comparing the degrees of \(P(x)\) and \(Q(x)\):

• If the degree of \(P(x)\) is less than that of \(Q(x)\), the horizontal asymptote is \(y = 0\).
• If the degrees are equal, as in our problem where both the numerator and denominator are linear, the asymptote is the ratio of their leading coefficients. For example, \(y = \frac{2}{5}\) in \(f(x) = \frac{2x + 3}{5x + 7}\).
• If the degree of \(P(x)\) is greater than that of \(Q(x)\), there is no horizontal asymptote (though there may be an oblique one).

In practice, horizontal asymptotes imply the long-term behavior of a function; regardless of how much \(f(x)\) changes, as \(x\) heads toward positive or negative infinity, it will level off towards this asymptote, like a bird gliding towards the horizon.

Polynomial Functions

Polynomial functions are a key part of various mathematical concepts and possess distinctive characteristics based on their varying forms and degrees. A polynomial is expressed as \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where each \(a_i\) is a coefficient and \(n\) is a non-negative integer representing the degree.
The degree of a polynomial indicates the number and nature of its roots as well as its end behavior - that is, how \(f(x)\) behaves as \(x\) approaches large positive or negative values. In rational functions, the polynomial division essentially guides us in identifying dominant terms to ascertain limits and asymptotic behavior.

• Lower degree terms become negligible as \(x\) grows large.
• The leading term (of the highest degree) governs the polynomial's rise or fall at extremes.

Thus, when analyzing a rational function like \( \frac{2x + 3}{5x + 7} \), both the numerator and the denominator are linear polynomials, meaning they dictate the rational function's limit and asymptotic nature as \(x\) trends towards infinity.