Problem 19 - Calculus Early Transcendentals

Question

15-36 Find the limit. $$ \lim _{x \rightarrow \infty} \frac{x^{3}+5 x}{2 x^{3}-x^{2}+4} $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Limit Expression

We are required to find the limit of the function \( \frac{x^{3}+5x}{2x^{3}-x^{2}+4} \) as \( x \) approaches infinity.

Step 2: Identify Dominant Terms

In the expression \( \frac{x^{3}+5x}{2x^{3}-x^{2}+4} \), the highest power of \( x \) in the numerator is \( x^3 \) and in the denominator is \( 2x^3 \). These terms will dominate the behavior of the function as \( x \) approaches infinity.

Step 3: Simplify the Expression

Divide every term in both the numerator and the denominator by \( x^3 \), the highest power of \( x \) in the expression: \[ \frac{\frac{x^{3}}{x^{3}} + \frac{5x}{x^{3}}}{\frac{2x^{3}}{x^{3}} - \frac{x^{2}}{x^{3}} + \frac{4}{x^{3}}} = \frac{1 + \frac{5}{x^2}}{2 - \frac{1}{x} + \frac{4}{x^3}} \].

Step 4: Evaluate the Limit by Substitution

As \( x \rightarrow \infty \), the fractions \( \frac{5}{x^2} \), \( \frac{1}{x} \), and \( \frac{4}{x^3} \) all approach zero. Substitute these limits into the equation: \[ \frac{1+0}{2-0+0} = \frac{1}{2} \].

Step 5: State the Result

The limit of the given function as \( x \rightarrow \infty \) is \( \frac{1}{2} \).

Key Concepts

Dominant Terms

In calculus, understanding dominant terms can significantly simplify finding limits of functions, especially as the input grows extremely large or small. Dominant terms are typically the highest power terms in polynomial functions. They are the ones that grow faster than any other terms as the function arguments increase or decrease.

For instance, in the expression \( x^3 + 5x \), the term \( x^3 \) is dominant because it will outgrow \( 5x \) significantly as \( x \) approaches infinity. Similarly, in the denominator \( 2x^3 - x^2 + 4 \), the term \( 2x^3 \) is dominant. This is important for simplifying the problem because in many cases, the lesser terms become negligible as \( x \) becomes very large.

To find limits involving infinity, always identify and focus on these dominant terms first. They will primarily determine the behavior of the function at the extreme values, making it easier to calculate the limit.

Rational Functions

Rational functions are fractions where both the numerator and the denominator are polynomials. The general form of a rational function is \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials. These functions can have intriguing behavior, especially as \( x \) approaches certain critical values like infinity or where the denominator goes to zero.

A key aspect in analyzing limits of rational functions at infinity is to compare their dominant terms. Understanding this is crucial because it simplifies the process by reducing the function to a simpler form, often a simple fraction. This step is what makes dominant terms helpful—they allow us to focus only on the terms that significantly affect the outcome of the limit.

Knowing how to handle these can give insights into the behavior of more complicated functions and is a strong tool in calculus.

Infinity

Infinity in calculus is not a number but a concept that represents unbounded growth or decay. It's a way to describe what happens when variables go beyond all bounds—either becoming very large positively or negatively. Calculus uses this idea to explore the behavior of functions in such "far-off" scenarios.

When dealing with limits, infinity allows us to consider what happens when a variable doesn't settle into a particular number but instead keeps growing. In the context of limits of rational functions, as \( x \) approaches infinity, the smaller terms in both numerator and denominator tend towards zero.

The use of infinity in limits essentially focuses on determining how the dominant terms behave, which helps simplify calculations. Recognizing infinity in calculus helps to encapsulate trends and patterns, offering a clearer insight into the structure of functions.