Problem 14 - Thomas Calculus

Question

In Exercises \(13-22,\) find the limit of each rational function (a) as \(x \rightarrow \infty\) and \((b)\) as \(x \rightarrow-\infty\) . $$f(x)=\frac{2 x^{3}+7}{x^{3}-x^{2}+x+7}$$

Step-by-Step Solution

Answer

The limit is 2 as both \(x \to \infty\) and \(x \to -\infty\).

Step 1: Identify the Leading Terms

Examine the given rational function: \(f(x) = \frac{2x^3 + 7}{x^3 - x^2 + x + 7}\). The leading term in the numerator is \(2x^3\) and in the denominator is \(x^3\). These terms dictate the behavior of the function as \(x\) approaches infinity or negative infinity.

Step 2: Simplify the Expression

To examine the limits as \(x\) approaches \(\infty\) or \(-\infty\), divide each term of the rational function by \(x^3\) (the highest power of \(x\) in the denominator): \[\frac{2x^3 + 7}{x^3 - x^2 + x + 7} = \frac{2 + \frac{7}{x^3}}{1 - \frac{1}{x} + \frac{1}{x^2} + \frac{7}{x^3}}.\] This approach simplifies the function and focuses on the leading terms.

Step 3: Evaluate the Limit as \(x \to \infty\)

As \(x\) approaches infinity, the terms with \(1/x\), \(1/x^2\), and \(1/x^3\) approach zero. Thus, the expression simplifies to \[\frac{2 + 0}{1 - 0 + 0 + 0} = 2.\] Therefore, the limit as \(x \rightarrow \infty\) is 2.

Step 4: Evaluate the Limit as \(x \to -\infty\)

Similarly, as \(x\) approaches negative infinity, the terms with \(1/x\), \(1/x^2\), and \(1/x^3\) again approach zero. Thus, the expression simplifies to \[\frac{2 + 0}{1 - 0 + 0 + 0} = 2.\] Therefore, the limit as \(x \rightarrow -\infty\) is also 2.

Key Concepts

Limits

Limits are fundamental in understanding how functions behave as the input values become very large or very small. In the context of rational functions, limits help us explore what happens to the function values as the variable approaches infinity or negative infinity. In general, the symbol \( \lim_{x \to a} f(x) \) indicates what value the function \( f(x) \) approaches as \( x \) gets closer to \( a \).

For rational functions like \( f(x) = \frac{2x^3 + 7}{x^3 - x^2 + x + 7} \), examining limits involves finding what happens when \( x \to \infty \) or \( x \to -\infty \).

• The limit as \( x \to \infty \) shows the behavior of the function when \( x \) takes on very large positive values.
• The limit as \( x \to -\infty \) assesses the behavior when \( x \) takes on very large negative values.

By simplifying the expression with respect to the leading terms, we can see that the limit remains stable and finite, which is key for understanding the function's end behavior.

Infinity

Infinity is not a number but rather a concept used to describe something that is unbounded or limitless. When examining the behavior of functions as \( x \) approaches infinity (\( x \to \infty \)) or negative infinity (\( x \to -\infty \)), we are essentially exploring the end behavior of the function.

In mathematical analysis, we use infinity to analyze:

• The end behavior of functions as their input grows without bound.
• How functions respond in extremely large or small contexts.
• The trends or stability in functions' outputs as inputs stretch beyond any finite limit.

In the given rational function example, analyzing the limit as \( x \to \infty \) or \(-\infty \) lets us see if the function stabilizes or diverges. The rational function \( \frac{2x^3 + 7}{x^3 - x^2 + x + 7} \), simplifies significantly due to these limits, pointing out that some parts of the function become negligible when \( x \) is infinitely large or small.

Leading Term

The leading term in a polynomial is the term with the highest power of the variable. In rational functions, identifying the leading terms of both the numerator and the denominator is crucial, as these terms determine the behavior of the function as \( x \) approaches infinity or negative infinity.

Think of a rational function \( f(x) = \frac{2x^3 + 7}{x^3 - x^2 + x + 7} \):

• The leading term in the numerator is \( 2x^3 \).
• The leading term in the denominator is \( x^3 \).

By focusing on these terms, you simplify the analysis of limits. As \( x \) becomes very large, the lower power terms (like \( x^2, x, \text{and constants} \)) contribute less and less, becoming negligible in comparison to the leading terms. This simplification allows us to understand that the function's behavior, as \( x \to \infty \) or \(-\infty \), is predominantly impacted by the leading terms, producing a stable limit of 2 in our example.