Problem 9 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the limits. \(\lim _{x \rightarrow \infty} \frac{3 x^{3}-x^{2}}{\pi x^{3}-5 x^{2}}\)

Step-by-Step Solution

Answer

The limit is \(\frac{3}{\pi}\).

Step 1: Understand the Function

We are given the function \(f(x) = \frac{3x^3 - x^2}{\pi x^3 - 5x^2}\) and we need to find the limit as \(x\) approaches infinity.

Step 2: Identify the Dominating Term

For both the numerator \(3x^3 - x^2\) and the denominator \(\pi x^3 - 5x^2\), the highest degree term is \(x^3\). These are the dominating terms in their respective expressions.

Step 3: Simplify the Expression

Divide every term by \(x^3\), the highest power of \(x\) present in the function. This gives us: \[\lim_{x \rightarrow \infty} \frac{3x^3 - x^2}{\pi x^3 - 5x^2} = \lim_{x \rightarrow \infty} \frac{\frac{3x^3}{x^3} - \frac{x^2}{x^3}}{\frac{\pi x^3}{x^3} - \frac{5x^2}{x^3}} = \lim_{x \rightarrow \infty} \frac{3 - \frac{1}{x}}{\pi - \frac{5}{x}}\]

Step 4: Evaluate the Limit

As \(x\) approaches infinity, \(\frac{1}{x}\) and \(\frac{5}{x}\) both approach 0. The expression therefore simplifies to: \[\frac{3 - 0}{\pi - 0} = \frac{3}{\pi}.\]

Key Concepts

Dominating Terms

In calculus, when you are dealing with functions that grow very large, it's important to identify the 'dominating terms'. These terms essentially determine the behavior of the function as variables tend towards infinity. **Dominating terms** are the highest power terms in a polynomial.

For example, consider the function:

• Numerator: \(3x^3 - x^2\)
• Denominator: \(\pi x^3 - 5x^2\)

In both the numerator and denominator, \(x^3\) is the dominating term because it has the highest power. As \(x\) becomes very large, the influence of lower power terms like \(x^2\) diminishes greatly in comparison to the \(x^3\) term.

By focusing on these dominating terms, we can simplify the process of evaluating limits since they show which parts of the function grow the fastest, and will louder the less impactful terms.

Rational Functions

Rational functions are fractions where both the numerator and the denominator are polynomials. In the context of limits, particularly when \(x\) approaches infinity, these functions can be challenging

• The given expression \(\frac{3x^3 - x^2}{\pi x^3 - 5x^2}\) is a rational function.
• The degrees of the polynomials (highest power of \(x\)) in the numerator and denominator are both 3, directly affecting how these types of functions simplify as \(x\) tends towards infinity.

When evaluating such functions, understanding the relation between the degrees of these polynomials is crucial. Different degrees can lead to different types of limits:

• If the degree of the numerator is greater than the degree of the denominator, the limit will tend to infinity.
• If the degree of the numerator is less than the degree of the denominator, the limit will tend to zero.
• If the degrees are equal, like in our example, the limit will be the ratio of the leading coefficients of the dominating terms, which is \(\frac{3}{\pi}\) in this exercise.

Infinity Limits

Infinity limits refer to finding the behavior of a function as the input grows beyond bounds, often to infinity. In calculus, these limits are especially important for rational functions.

The method to solve involves simplifying the function so that it's easier to evaluate. Typically, this involves dividing all terms by the highest power of \(x\) in the denominator. Here's how it works in our case:

• Start with: \(\lim_{x \rightarrow \infty} \frac{3x^3 - x^2}{\pi x^3 - 5x^2}\)
• Divide every term by \(x^3\) to get: \(\frac{3 - \frac{1}{x}}{\pi - \frac{5}{x}}\)
• As \(x\) moves towards infinity, \(\frac{1}{x}\) and \(\frac{5}{x}\) approach 0.
• Thus, the expression simplifies to \(\frac{3}{\pi}\).

This reveals how functions behave at extreme values, which is a fundamental concept in understanding calculus and the limits of functions as variables move towards infinity.