Problem 17 - Calculus: Early Transcendentals

Question

Find the limit or show that it does not exist. \( \displaystyle \lim_{x \to -\infty}\frac{x - 2}{x^2 + 1} \)

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Identify the Form of the Limit

We need to find the limit of the rational function \( \frac{x - 2}{x^2 + 1} \) as \( x \to -\infty \). This involves analyzing the behavior of both the numerator and the denominator as \( x \) approaches -\infty.

Step 2: Determine Dominant Terms

As \( x \to -\infty \), the dominant term in the numerator \( x - 2 \) is \( x \). Similarly, the dominant term in the denominator \( x^2 + 1 \) is \( x^2 \).

Step 3: Simplify the Expression Using Dominant Terms

The limit simplifies to:\[ \lim_{x \to -\infty} \frac{x - 2}{x^2 + 1} \approx \lim_{x \to -\infty} \frac{x}{x^2} = \lim_{x \to -\infty} \frac{1}{x} \]As the \( x \to -\infty \).

Step 4: Analyze the Simplified Limit

The limit \( \lim_{x \to -\infty} \frac{1}{x} \) approaches 0 because as \( x \) becomes very large (negatively), \( \frac{1}{x} \) becomes closer and closer to 0.

Step 5: Conclusion on the Limit

By following the steps, we determined that the limit of \( \frac{x - 2}{x^2 + 1} \) as \( x \to -\infty \) is 0.

Key Concepts

Rational Functions

Rational functions are a type of function represented by the ratio of two polynomials. The general form is \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials. These functions are important in calculus because they can exhibit interesting behavior at various points, such as asymptotic behavior or holes where the function is undefined.

To work with a rational function effectively, it’s crucial to understand the roles of the numerator and denominator:

• The numerator, \( P(x) \), determines the values that the function can reach.
• The denominator, \( Q(x) \), impacts where the function is undefined (where \( Q(x) = 0 \)).

When analyzing rational functions, especially in the context of limits, it is helpful to identify the dominant terms in both the numerator and the denominator. Doing so allows you to simplify the function by focusing on the terms that most significantly affect the function’s behavior, particularly as \( x \) approaches extreme values like \( \pm \infty \).

Limit at Infinity

Limits at infinity are used to determine how a function behaves as the input, \( x \), becomes very large, whether positively or negatively. This is particularly relevant for rational functions, which often involve terms with different powers of \( x \) that dominate at different values.

When evaluating limits at infinity for rational functions:

• Examine the degree of the polynomials in the numerator and the denominator.
• The degree is the highest power of \( x \) in the polynomial expression.

If the degree of the polynomial in the numerator is less than that in the denominator, as in our example, the limit is zero. Conversely, if the degree in the numerator is greater, the limit is infinity or negative infinity, depending on the sign of the leading terms. If both have the same degree, the limit is the ratio of their leading coefficients. Understanding these rules makes it easier to quickly determine the behavior of rational functions as \( x \to \pm \infty \).

Dominant Terms

Dominant terms are the parts of a mathematical expression that have the most significant impact on its value as \( x \) becomes very large or very small. In rational functions, identifying the dominant terms is a key step in simplifying expressions before calculating limits, especially at infinity.

Consider a rational function \( \frac{x - 2}{x^2 + 1} \):

• In the numerator, \( x \) is the dominant term because it outweighs \( -2 \) as \( x \) grows large.
• In the denominator, \( x^2 \) takes precedence over \( +1 \).

By focusing on the dominant terms, the expression simplifies to \( \frac{x}{x^2} \), which can be further reduced to \( \frac{1}{x} \). Understanding dominant terms allows us to predict the behavior of functions and solve limits more efficiently. This concept is essential for dealing with complex functions in calculus, where simplifying problems can lead to powerful insights and solutions.