Problem 21 - Calculus of a Single Variable

Question

Finding a Limit In Exercises \(19-38,\) find the limit. $$ \lim _{x \rightarrow \infty} \frac{2 x-1}{3 x+2} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the highest power of x in the denominator

In the denominator, the highest power of \(x\) is 1, because the denominator is \(3x + 2\).

Step 2: Divide each term in the rational function by \(x\) to the highest power in the denominator

This will give \(\frac{2-\frac{1}{x}}{3+\frac{2}{x}}\). As \(x\) approaches infinity, \( \frac{1}{x} \) and \( \frac{2}{x} \) both approach zero.

Step 3: Find the limit

Taking the limit as \(x\) approaches infinity for will give \(\lim_{x \rightarrow \infty} \frac{2-\frac{1}{x}}{3+\frac{2}{x}} = \frac{2-0}{3-0} = \frac{2}{3}.\)

Key Concepts

Limit at Infinity

When we talk about the "limit at infinity," we are interested in understanding the behavior of a function as the variable approaches infinity. This is particularly important in calculus as it helps to establish long-term behavior or trends of a function. For example, if we want to determine the limit of a rational function as the variable goes to infinity, we are trying to see what value (if any) the function approaches. In the exercise you provided, \[\lim_{x \rightarrow \infty} \frac{2x-1}{3x+2}\]is calculated by observing the behavior of the fractions when \( x \) is extremely large.

• As \( x \) approaches infinity, smaller terms like \(-1\) or \(+2\) become insignificant compared to terms with \(x\).
• The function is simplified by dividing each term by the highest power of \(x\) present in the denominator.
• Once simplified, the limit of each term is observed as \( x \) reaches infinity, often leading to terms like \(\frac{1}{x}\) and \(\frac{2}{x}\) approaching zero.

Understanding this process is crucial as it helps simplify complex functions and identify their limits.

Rational Functions

Rational functions are quotients of polynomials. They take the form \(\frac{P(x)}{Q(x)}\) where \(P(x)\) and \(Q(x)\) are polynomials. In the context of calculus, they are significant because their limits can tell us about the behavior of the entire function.

• The degree of the numerator and the denominator often determines the limit behavior at infinity.
• For example, if the degree of the numerator is equal to the degree of the denominator, the limit at infinity is the ratio of their leading coefficients.
• If the degree of the numerator is less than the degree of the denominator, the limit at infinity is 0.
• If the degree of the numerator is greater than the degree of the denominator, the limit does not exist in a traditional sense and may be influenced by the direction of infinity.

In the provided problem \(\frac{2x-1}{3x+2}\), both the numerator and denominator are degree 1, leading us to evaluate the ratio of the coefficients \(\frac{2}{3}\). Rational functions help us understand complex polynomial behaviors by focusing on long-term trends and behaviors.

Asymptotic Behavior

Asymptotic behavior refers to how a function behaves as its inputs become very large (positively or negatively). This concept includes understanding horizontal and vertical asymptotes, which are lines that the graph of a function approaches but never touches. In the case of rational functions, horizontal asymptotes are of particular interest.

• Horizontal asymptotes occur when evaluating the limit at infinity, showing how the function stabilizes or levels off.
• In problems like \(\frac{2x-1}{3x+2}\), understanding asymptotic behavior helps in predicting the graph's shape and its end behavior.

Horizontal asymptotes are often direct results from limits at infinity. For rational functions, if the limit conclusively is a real number as \( x \) approaches infinity, a horizontal asymptote exists at \(y = \text{(that real number)}\). In the example, the limit found was \(\frac{2}{3}\), meaning there is a horizontal asymptote at \(y = \frac{2}{3}\). Knowing asymptotic behavior is helpful for sketching graphs and in practical applications where predictions of long-term trends are essential.