Problem 13 - Calculus: Early Transcendentals

Question

Evaluate the limit and justify each step by indicating the appropriate properties of limits. \( \displaystyle \lim_{x \to \infty} \dfrac{2x^2 - 7}{5x^2 + x -3} \)

Step-by-Step Solution

Answer

The limit is \(\frac{2}{5}\).

Step 1: Identify the Polynomial Degree

Examine the degrees of the polynomial in the numerator and the denominator. The highest degree term in the numerator is \(2x^2\) (degree 2), and the highest degree term in the denominator is \(5x^2\) (degree 2). Both have the same degree.

Step 2: Simplify by Factoring Out Highest Degree Terms

To simplify the limit, factor out the highest degree term from both the numerator and the denominator. In this case, factor out \(x^2\) from the numerator and \(x^2\) from the denominator:\[\lim_{x \to \infty} \frac{2x^2 - 7}{5x^2 + x - 3} = \lim_{x \to \infty} \frac{x^2(2 - \frac{7}{x^2})}{x^2(5 + \frac{1}{x} - \frac{3}{x^2})}\]

Step 3: Cancel Common Factor

Cancel the common factor \(x^2\) from both the numerator and the denominator:\[\lim_{x \to \infty} \frac{2 - \frac{7}{x^2}}{5 + \frac{1}{x} - \frac{3}{x^2}}\]

Step 4: Evaluate Limit by Substitution

As \(x\) approaches infinity, the terms \(-\frac{7}{x^2}\), \(\frac{1}{x}\), and \(-\frac{3}{x^2}\) approach zero. Thus, the limit simplifies to:\[\frac{2}{5}\]

Step 5: Conclusion

Using the limits of rational functions, if the degree of the numerator and the degree of the denominator are the same, the limit equals the ratio of the leading coefficients (here 2 and 5). Therefore, the result of the limit is \(\frac{2}{5}\).

Key Concepts

Polynomial Functions

Polynomial functions are expressions containing variables raised to non-negative integer powers. A key feature of polynomial functions is their degree, which is the highest power of the variable in the expression. For example, in the function \(2x^2 - 7\), the degree is 2 because the highest power is \(x^2\).

Polynomial functions can have various forms:

• Linear polynomials (degree 1): e.g., \(3x + 2\)
• Quadratic polynomials (degree 2): e.g., \(x^2 - 4x + 4\)
• Cubic polynomials (degree 3): e.g., \(x^3 - x^2 + x - 5\)

The degree of the polynomial affects its graph, indicating how the function behaves as \(x\) becomes very large or very small. In limits, identifying the degree helps in determining the function's behavior at infinity.

Rational Functions

Rational functions are ratios of two polynomial functions. They have the form \(\frac{P(x)}{Q(x)}\), where both \(P(x)\) and \(Q(x)\) are polynomials. The example given, \(\frac{2x^2 - 7}{5x^2 + x - 3}\), is a rational function.

When evaluating limits of rational functions as \(x\) approaches infinity, it is essential to consider:

• The degrees of the numerator and the denominator polynomials.
• Any simplifications that can be applied, such as factoring out common terms.

These functions are particularly interesting to study at infinity because their asymptotic behavior heavily relies on the polynomial degrees. A rational function can have horizontal, vertical, or no asymptotes, which are often studied using limits.

Infinite Limits

Infinite limits involve finding the behavior of a function as the variable either goes to infinity or negative infinity. In the exercise, the aim was to find \(\lim_{x \to \infty} \frac{2x^2 - 7}{5x^2 + x -3}\).

To tackle infinite limits for rational functions, look at the degrees of the polynomials involved:

• If the numerator's degree is higher, the function approaches infinity.
• If the denominator's degree is higher, the function approaches zero.
• If both degrees are equal, the limit is the ratio of the leading coefficients.

In our case, both the polynomials have degree 2. Thus, the limit will equal the ratio of the leading coefficients from both polynomials, resulting in \(\frac{2}{5}\).

Limit Properties

Limits have several useful properties that help us solve problems involving limits in calculus. These properties can simplify complex problems and offer insights into function behavior. Here are a few crucial properties:

• The limit of a sum equals the sum of the limits: \(\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)\).
• The limit of a product equals the product of the limits: \(\lim_{x \to a} [f(x) g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)\).
• For rational functions where the degrees of the numerator and denominator are equal, the limit at infinity is the ratio of the leading coefficients if the limits exist.

These properties are crucial when analyzing functions as they approach specific points or infinity, and they make evaluating the original problem possible. By employing these properties, it becomes easier to simplify the expression and determine that \(\frac{2}{5}\) is the limit as \(x\) approaches infinity.