Problem 18 - Thomas Calculus

Question

Show that \(\sqrt{x^{4}+x}\) and \(\sqrt{x^{4}-x^{3}}\) grow at the same rate as \(x \rightarrow \infty\) by showing that they both grow at the same rate as \(x^{2}\) as \(x \rightarrow \infty\)

Step-by-Step Solution

Answer

Both functions grow at the rate of \(x^2\) as \(x \to \infty\).

Step 1: Simplify the Expression for Large x

Consider \(\sqrt{x^4 + x}\) as \(x \to \infty\), the dominant term inside the square root is \(x^4\). So, \(\sqrt{x^4 + x}\) simplifies to \(\sqrt{x^4(1 + \frac{x}{x^4})} = x^2\sqrt{1 + \frac{1}{x^3}}\). As \(x \to \infty\), the term \(\frac{1}{x^3}\) approaches zero, hence it simplifies to \(x^2\).

Step 2: Simplify the Other Expression for Large x

Now consider \(\sqrt{x^4 - x^3}\). Similarly, the dominant term \(x^4\) gives \(\sqrt{x^4(1 - \frac{x^3}{x^4})} = x^2\sqrt{1 - \frac{1}{x}}\). As \(x \to \infty\), \(\frac{1}{x}\) tends to zero, leaving us with \(x^2\).

Step 3: Comparison to x^2

Both expressions simplify to \(x^2\) as \(x \to \infty\). This means that both \(\sqrt{x^4 + x}\) and \(\sqrt{x^4 - x^3}\) grow at the same rate as \(x^2\), confirming that they grow at the same rate as each other as \(x \to \infty\).

Key Concepts

Dominant Term

When working with complex functions, especially for very large values of the variable, it's useful to determine the dominant term inside an expression. The dominant term is the part of the expression that grows the fastest and has the greatest impact on the function's behavior for large values. In this context, it is especially relevant for simplification and asymptotic analysis. Consider the expression \( \sqrt{x^4 + x} \). As \( x \to \infty \), while both terms inside the square root, \( x^4 \) and \( x \), grow, \( x^4 \) grows much faster than \( x \). Therefore, \( x^4 \) is the dominant term.

• The expression simplifies to \( \sqrt{x^4(1 + \frac{1}{x^3})} \) by factoring out \( x^4 \).
• As \( x \) becomes very large, \( \frac{1}{x^3} \) becomes negligible, simplifying the square root to \( x^2 \).

This insight allows us to focus on the most significant part of the expression, ignoring smaller terms that have less impact as \( x \to \infty \). Similarly, for \( \sqrt{x^4 - x^3} \), \( x^4 \) remains the dominant term:

• It simplifies to \( \sqrt{x^4(1 - \frac{1}{x})} \).
• Since \( \frac{1}{x} \to 0 \), the square root approaches \( x^2 \).

Understanding the dominant term helps us to see that both expressions behave similarly for large \( x \). This makes further analysis more manageable.

Square Root Simplification

Square root simplification is a critical tool used in analyzing functions, especially within asymptotic analysis. By simplifying the square root, we can extract useful information about the growth behavior of functions as variables become large, typically represented by \( x \to \infty \). This step focuses on reducing complex expressions to forms that are easier to compare and interpret.Given the expression \( \sqrt{x^4 + x} \), the simplification process involves understanding that the terms within the square root can be split.

• For \( x^4 + x \), factoring out \( x^4 \) gives \( \sqrt{x^4(1 + \frac{x}{x^4})} = x^2\sqrt{1 + \frac{1}{x^3}} \).
• As \( x \) increases, \( \frac{1}{x^3} \to 0 \), making the radical approximate \( x^2 \).

Similarly, for \( \sqrt{x^4 - x^3} \):

• It simplifies as \( \sqrt{x^4(1 - \frac{x^3}{x^4})} = x^2\sqrt{1 - \frac{1}{x}} \).
• When \( x \) is large, \( \frac{1}{x} \) becomes negligible, leading again to \( x^2 \).

The square root simplification shows us that despite initial different appearances, the terms under consideration both approximate the same simple function when \( x \to \infty \). This approach reduces complexity, enabling a more straightforward comparison.

Growth Rate Comparison

In the realm of asymptotic analysis, comparing growth rates is essential for understanding how different functions behave as variables tend toward infinity. The objective is to determine whether two or more functions grow at the same speed by observing their simplifications and dominant terms.For functions \( \sqrt{x^4 + x} \) and \( \sqrt{x^4 - x^3} \), after identifying the dominant term and simplifying the square roots, we noted that both expressions simplify to \( x^2 \) as \( x \to \infty \). Therefore, even if their full expressions differ, their growth rates are identical.

• The simplification of each function to \( x^2 \) establishes that, asymptotically, they can be treated equivalently in terms of growth rate.
• Thus, both \( \sqrt{x^4 + x} \) and \( \sqrt{x^4 - x^3} \) grow at the same rate as \( x^2 \).

This demonstrates that complex-looking functions might share the same asymptotic behavior, which can be particularly useful in calculus and algorithm analysis, where comparing growth rates simplifies understanding and predictions. By focusing on what happens as \( x \to \infty \), we can often predict performance and make informed decisions about which functions will dominate or behave similarly in practical scenarios.