91Ó°ÊÓ

When it comes to calculus, rational functions are expressions that involve ratios of polynomials. A typical form looks like \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials. Understanding their limits, especially as \( x \) approaches infinity, can tell us a lot about their long-term behavior.

When you have a rational function and you want to determine its limit as \( x \to \infty \), the best first step is to simplify it, often by dividing each term in the numerator and the denominator by the highest power of \( x \) present in the denominator. This makes the expression easier to handle, particularly for large values of \( x \). In our example, we divided each term by \( x \) because the highest power of \( x \) in the denominator was \( x \) itself.

Another useful tip is to closely observe the leading coefficients of the highest power terms. After simplification, they usually govern the limit of the entire function. In our case, the rational function simplifies to \( \frac{2 - \frac{1}{x}}{3 + \frac{2}{x}} \), leading to the limit \( \frac{2}{3} \).

Properties of limits make evaluating limits significantly easier. They help break down complex functions into more manageable parts. Some key properties include:

• If \( \lim_{x \to c} f(x) = L \) and \( \lim_{x \to c} g(x) = M \), then \( \lim_{x \to c} [f(x) + g(x)] = L + M \).
• If \( \lim_{x \to c} f(x) = L \), then \( \lim_{x \to c} cf(x) = cL \) for any constant \( c \).
• \( \lim_{x \to c} [f(x) \cdot g(x)] = L \cdot M \).
• \( \lim_{x \to c} \left( \frac{f(x)}{g(x)} \right) = \frac{L}{M} \) provided \( M eq 0 \).

In our exercise, the terms \( \frac{1}{x} \) and \( \frac{2}{x} \) vanish to zero as \( x \) approaches infinity. This is a direct application of the limit properties, making the evaluation straightforward. By recognizing these small terms approaching zero, we are left with the simplified fraction \( \frac{2}{3} \).

Asymptotic behavior helps us understand how functions act at the extremes, particularly as \( x \) approaches infinity or negative infinity. It's a crucial concept in determining the limits at infinity for rational functions.

For a rational function like \( \frac{2x-1}{3x+2} \), its asymptotic behavior at infinity can be determined by observing the powers of \( x \) in the numerator and the denominator. If the degrees (the highest powers of \( x \)) are the same, the horizontal asymptote is simply the ratio of the leading coefficients, which is \( \frac{2}{3} \) in this case.

Understanding this leads to two key insights:

• If the degree of the polynomial in the numerator is less than that in the denominator, the horizontal asymptote is \( y = 0 \).
• If the degree of the numerator is greater, there is no horizontal asymptote, and the growth is unbounded.

Thus, identifying the degrees and leading coefficients will often directly give the asymptotic behavior of the function, simplifying the process of finding the limit.