Problem 15 - Essential Calculus

Question

In Exercises \(15-36,\) find the limit. $$ \lim _{x \rightarrow \infty} \frac{2 x-1}{3 x+2} $$

Step-by-Step Solution

Answer

The limit of the function \( \frac{2x-1}{3x+2} \) as \( x \) approaches infinity is \( \frac{2}{3} \).

Step 1: Simplify the function

To make the function simpler, divide every term in the numerator and the denominator by \( x \). The function becomes: \( \frac{\frac{2x}{x}-\frac{1}{x}}{\frac{3x}{x}+\frac{2}{x}} \). After simplifying, the function becomes: \( \frac{2-\frac{1}{x}}{3+\frac{2}{x}} \).

Step 2: Taking limit as x approaches infinity

Knowing that \( \frac{1}{x} \) will approach 0 as \( x \) approaches infinity, \( \frac{2-\frac{1}{x}}{3+\frac{2}{x}} \) can be simplified to \( \frac{2-0}{3+0} \).

Step 3: Simplify the equations

After simplifying \( \frac{2-0}{3+0} \), you will get \( \frac{2}{3} \). Therefore, \( \lim _{x \rightarrow \infty} \frac{2 x-1}{3 x+2} = \frac{2}{3} \)

Key Concepts

Rational Functions

Rational functions are mathematical expressions represented as the ratio of two polynomials. They are common in algebra and calculus, and they can take many different forms, such as \( \frac{f(x)}{g(x)} \), where \( f(x) \) and \( g(x) \) are polynomial functions.

Here are some basic properties of rational functions:

• They can be simplified like fractions by factoring the numerator and the denominator.
• They often have asymptotes, which are lines that the graph approaches but never touches.

In our exercise, the function is \( \frac{2x - 1}{3x + 2} \). Here, both the numerator and denominator are linear polynomials in \( x \). To simplify such functions, a common method is to divide every term by \( x \). This helps us to reveal their behavior as \( x \) approaches infinity or negative infinity.

It's important to identify the leading terms, which dominate the behavior of the function at both ends of the number line. In this exercise, these were the terms \( 2x \) and \( 3x \) from the numerator and denominator, respectively.

Asymptotic Behavior

Asymptotic behavior helps us understand what happens to a function as the variable approaches infinity or a specific point. For rational functions, this often involves considering the limiting behavior near vertical asymptotes (which occur where the denominator is zero) or horizontal asymptotes (which are determined by the degrees of the numerator and denominator polynomials).

In our exercise, the solution focused on what happens as \( x \) approaches infinity. We simplified the function to \( \frac{2 - \frac{1}{x}}{3 + \frac{2}{x}} \). As \( x \) becomes infinitely large, terms like \( \frac{1}{x} \) and \( \frac{2}{x} \) tend to zero. This leaves us with \( \frac{2}{3} \) as the horizontal asymptote.

This is characteristic of rational functions where the degrees of the numerator and denominator are equal. The coefficients of the leading terms (those with the highest degree) determine the horizontal asymptote. As a guideline:

• If the degree of the numerator is higher, the function increases or decreases toward infinity.
• If the degree of the numerator is less, the function approaches zero.
• If they are equal, the function approaches the ratio of the leading coefficients.

Infinity

The concept of infinity is crucial in calculus, especially when discussing limits. Infinity (\( \infty \)) is not a number but an idea that describes something without bound. When we say \( x \to \infty \), we're discussing what happens as \( x \) grows larger and larger without stopping.

In evaluating \( \lim_{x \to \infty} \frac{2x - 1}{3x + 2} \), our task is to determine the value that the function approaches as \( x \) goes to infinity. This is done by finding limits. When \( x \) becomes very large, parts of the function that diminish (like \( \frac{1}{x} \)) become negligible, simplifying our calculations.

In a practical sense, understanding limits as \( x \to \infty \) helps us predict how functions behave in extreme scenarios, which is valuable across many fields such as physics and engineering.