Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the limits. \(\lim _{x \rightarrow \infty} \frac{x^{3}}{2 x^{3}-100 x^{2}}\)

Step-by-Step Solution

Answer

The limit is \( \frac{1}{2} \).

Step 1: Identify the highest degree term

In the expression \( \frac{x^{3}}{2x^{3} - 100x^{2}} \), identify the highest degree term in both the numerator and the denominator. Here, both have \( x^3 \) as the highest degree term.

Step 2: Simplify the expression

To simplify the limit, divide every term in the numerator and the denominator by \( x^3 \), the highest degree term. This gives: \( \frac{\frac{x^{3}}{x^{3}}}{\frac{2x^{3}}{x^{3}} - \frac{100x^{2}}{x^{3}}} = \frac{1}{2 - \frac{100}{x}} \).

Step 3: Evaluate the limit

As \( x \) approaches infinity, \( \frac{100}{x} \) approaches zero. Therefore, the expression simplifies to \( \frac{1}{2} \) as \( x \to \infty \).

Key Concepts

Limit Calculation

When calculating limits, we aim to find the value that a function tends toward as the input approaches a specific point. In this case, we investigate what happens to \[ \lim _{x \rightarrow \infty} \frac{x^{3}}{2x^{3} - 100x^{2}} \]as \(x\) becomes infinitely large.

Here are some basic steps typically involved in limit calculation:

• Identify the terms contributing most significantly as \(x\) increases or decreases.
• Simplify the expression wherever possible to make it easier to evaluate the limit.
• Evaluate what the terms approach as \(x\) reaches the desired point (like infinity in this exercise).

Understanding these steps helps a lot in simplifying complex expressions and getting the desirable result, much like we simplified the given function to eventually find that the limit is \(\frac{1}{2}\). This process becomes even easier with more practice!

Rational Functions

Rational functions are expressions comprised of polynomials in both the numerator and the denominator, such as \[ \frac{x^3}{2x^3 - 100x^2} \].

Handling rational functions often involves recognizing the numerator and the denominator's impact on how the function behaves as \(x\) approaches infinity. To work with rational functions, consider:

• Analyzing the degree of the polynomials involved to determine the highest differential rate as \(x\) becomes large
• Using algebraic simplification techniques, such as factoring or dividing out terms to simplify evaluation

Rational functions can often be unruly at first glance, but by breaking them down into simpler parts and emphasizing the dominant terms, they become much more manageable. Once the setup is complete, you can quickly evaluate their behavior as \(x\) reaches extreme values.

Degree of Polynomial Terms

The degree of a polynomial term refers to the exponent of its variable. For example, in the term \(x^3\), the degree is 3.

This point is exceptionally important in rational functions, especially when evaluating limits at infinity. Here’s why:

• The highest degree term in a polynomial often dominates the behavior of the function as \(x\) approaches infinity. Thus, recognizing these terms is crucial when simplifying rational functions.
• Understanding the degree of terms helps in deciding which terms in the expression shrink towards zero or grow larger as \(x\) increases.
• In our exercise, knowing that both the numerator and denominator had a highest-degree term of \(x^3\) informed us that their growth was comparable, simplifying the evaluation.

By focusing primarily on the terms with the highest degree, especially as \(x\) approaches infinity, you can simplify expressions effectively and swiftly solve limit problems.