Problem 17 - Essential Calculus

Question

In Exercises \(15-36,\) find the limit. $$ \lim _{x \rightarrow \infty} \frac{x}{x^{2}-1} $$

Step-by-Step Solution

Answer

Therefore, \( \lim _{x \rightarrow \infty} \frac{x}{x^{2}-1} = 0 \)

Step 1: Identify the Limit

Here, we need to evaluate \( \lim _{x \rightarrow \infty} \frac{x}{x^{2}-1} \)

Step 2: Analyze the Function Type

This function is a simple fraction where both numerator and denominator are polynomials. This implies that it's a rational function.

Step 3: Apply the Limit Rule for Rational Functions

The rule states that: if the degree of the denominator is larger than the degree of the numerator, the limit of the function as x approaches infinity is 0.

Step 4: Find the Degrees of the Numerator and the Denominator

Here we see that the degree of the numerator is 1 and the degree of the denominator is 2. To find the degree of a polynomial, look for the highest exponent in the polynomial.

Step 5: Compare the Degrees

As degree of the denominator is higher than the degree of the numerator, according to the rule for rational functions, this limit is 0.

Key Concepts

Rational Functions

The degree of a polynomial is immensely important as it directly influences the shape and characteristics of its graph. In technical terms, the degree of a polynomial is the highest power (exponent) of the variable in a polynomial expression. It gives us immediate information about the potential number of roots (solutions) and the end behavior of the polynomial's graph.

For instance, the polynomial \(x^2 - 1\) has a degree of 2, because the highest power of \(x\) is 2. Knowing the degree of polynomials in both the numerator and the denominator is crucial when you’re dealing with rational functions and their limits at infinity. The degrees help us predict how fast each part of the function grows as \(x\) gets larger or smaller, enabling us to determine the limit of the function at infinity with greater ease.

Limits at Infinity

The concept of limits at infinity refers to the behavior of a function as the variable approaches either positive or negative infinity. In essence, it explores the question: 'What value does the function approach as the input grows without bound?' This concept is particularly integral when examining rational functions.

To determine the limit of a rational function as \(x\) approaches infinity, a key rule is used. If the polynomial degree of the denominator is higher than that of the numerator, the limit will be 0. This is because the denominator grows much faster than the numerator, rendering the values of the fraction closer and closer to zero as \(x\) increases. Likewise, if the degree of the numerator is higher, the limit of the function does not exist because it will grow towards positive or negative infinity.

Our example \(\lim _{x \rightarrow \infty} \frac{x}{x^{2}-1}\) illustrates this principle. The degree of the denominator (2) is greater than the degree of the numerator (1), leading us to conclude that the limit of this function as \(x\) approaches infinity is indeed 0. This overarching understanding of limits at infinity is crucial for mastering many aspects of mathematical analysis and calculus.