Problem 88 - Calculus and its applications

Question

Use GRAPH and TRACE to find each limit. When necessary, state that the limit does not exist. $$ \begin{array}{l} \text { For } g(x)=\frac{20 x^{2}}{x^{3}+2 x^{2}+5 x} \\ \text { find } \lim _{x \rightarrow \infty} g(x) \text { and } \lim _{x \rightarrow-\infty} g(x) . \end{array} $$

Step-by-Step Solution

Answer

Both limits, \( \lim_{x \to \infty} g(x) \) and \( \lim_{x \to -\infty} g(x) \), equal 0.

Step 1: Analyze the Function

The function given is \( g(x) = \frac{20x^2}{x^3 + 2x^2 + 5x} \). This is a rational function where the numerator is a quadratic polynomial \(20x^2\) and the denominator is a cubic polynomial \(x^3 + 2x^2 + 5x\). The degrees of the numerator and denominator are different (degree 2 and degree 3, respectively).

Step 2: Determine Behavior at Infinity

For \( x \rightarrow \infty \) or \( x \rightarrow -\infty \), the dominant terms (highest degree) dictate the limit. So we consider \( \frac{20x^2}{x^3} = \frac{20}{x} \).

Step 3: Evaluate Limit as x Approaches Infinity

As \( x \rightarrow \infty \), \( \frac{20}{x} \rightarrow 0 \) since \( 20/x \) becomes smaller as \( x \) increases. Thus the limit is \( \lim_{x \to \infty} g(x) = 0 \).

Step 4: Evaluate Limit as x Approaches Negative Infinity

Similarly, as \( x \rightarrow -\infty \), \( \frac{20}{x} \rightarrow 0 \) because \( 20/x \) still decreases in magnitude (moves closer to zero). Thus, \( \lim_{x \to -\infty} g(x) = 0 \).

Key Concepts

Rational Functions

Rational functions are a specific type of functions that are the ratio of two polynomial functions. In general, a rational function is expressed as \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). The behavior of these functions is heavily influenced by the degrees of the numerator and the denominator polynomials.
- **Degree of a Polynomial**: The degree is the highest power of \( x \) in the polynomial. For example, \(20x^2\) is a quadratic polynomial (degree 2), whereas \(x^3 + 2x^2 + 5x\) is a cubic polynomial (degree 3).
- **Important Cases**: The behavior at infinity or asymptotic behavior of rational functions depends on the comparison between the degrees of these polynomials.- If the degree of \( P(x) \) is less than the degree of \( Q(x) \), as in this example, the rational function approaches 0 as \( x \rightarrow \pm\infty \).
- A rational function is undefined where the denominator equates to zero. This usually results in vertical asymptotes or holes in the graph.

Behavior at Infinity

Understanding the behavior of functions at infinity is crucial for comprehending limits and asymptotic behavior. When evaluating the limit of rational functions as \( x \rightarrow \pm\infty \), we look for terms that remain significant at extreme values. These are generally the terms with the highest degree, also known as dominant terms.
- **Dominant Term Rule**: For rational functions, when \( x \rightarrow \pm\infty \), the highest powers in the numerator and the denominator determine the behavior. For example, for \( g(x) = \frac{20x^2}{x^3 + 2x^2 + 5x} \), the dominant terms are \( 20x^2 \) and \( x^3 \). This simplifies to \( \frac{20}{x} \), which approaches 0 as \( x \rightarrow \pm\infty \).
- **Graphical Interpretation**: Graphs often reveal the end behavior, showing that as \( x \) grows larger in either direction, the function value tends to stabilize around a horizontal asymptote, which, in this case, is \( y = 0 \).
- **Practical Implications**: This consideration of behavior is vital when predicting the outcomes of complicated functions in engineering, economics, or natural sciences, where understanding limits at infinity helps predict long-term trends.

Polynomial Functions

Polynomial functions play a foundational role in defining and understanding rational functions, as they form both the numerator and denominator in a rational expression. A polynomial function \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \) is a sum of terms, each consisting of a coefficient (\( a_i \)) multiplied by a variable (\( x \)) raised to an integer power.
- **Degree of a Polynomial**: The degree is indicated by the highest power of \( x \). This degree dictates the function's growth rate and its general shape when graphed. Higher degree terms will overshadow lower-degree terms as \( x \) moves away from the origin.
- **Roots of Polynomials**: These are values of \( x \) that make the polynomial equal to zero. They are important in understanding the behavior of polynomials themselves and the rational functions they form.
- **Simplification through Factorization**: In some cases, breaking down polynomials into products of their factors simplifies rational expressions and reveals asymptotic behavior more clearly. This factorization can simplify limits and lead to easier manipulation in complex calculations.
Understanding these elements helps manage complex algebraic expressions and ensures identifying significant traits, such as behavior at infinity, becomes manageable.