Problem 18 - Thomas Calculus in SI Units

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 same rate as \( x^2 \) as \( x \to \infty \).

Step 1: Understand the Expression

We need to compare the growth of two functions: \( \sqrt{x^{4}+x} \) and \( \sqrt{x^{4}-x^{3}} \) as \( x \to \infty \). To show they grow at the same rate as \( x^2 \), we will simplify each expression to see if they resemble \( \sqrt{x^4} \) as \( x \) becomes very large.

Step 2: Simplify \( \sqrt{x^{4}+x} \) as \( x \to \infty \)

Recognize that as \( x \to \infty \), the term \( x^4 \) dominates the expression \( x^4 + x \). Thus, \( \sqrt{x^{4}+x} \approx \sqrt{x^4} = x^2 \).

Step 3: Simplify \( \sqrt{x^{4}-x^3} \) as \( x \to \infty \)

Similarly, in the expression \( x^4 - x^3 \), the term \( x^4 \) is the leading term as \( x \to \infty \). Therefore, \( \sqrt{x^{4}-x^3} \approx \sqrt{x^4} = x^2 \).

Step 4: Conclude that Both Functions Grow at the Same Rate as \( x^2 \)

Since both \( \sqrt{x^{4}+x} \) and \( \sqrt{x^{4}-x^3} \) simplify to \( x^2 \) as \( x \to \infty \), they both grow at the same rate as \( x^2 \). Thus, they grow at the same rate relative to each other as \( x \to \infty \).

Key Concepts

Limits and Infinity

When we discuss limits and infinity, we are often dealing with how functions behave as a variable—usually referred to as \(x\)—grows without bound. Essentially, this means we want to understand what happens when \(x\) becomes very, very large. In mathematical terms, we explore the idea of \(x \to \infty\).

For the given expressions \(\sqrt{x^{4}+x}\) and \(\sqrt{x^{4}-x^{3}}\), the focus lies on how they change as \(x\) increases indefinitely. By examining these limits, we can identify dominant terms that determine their growth rate, helping us deduce important aspects like whether they grow at the same rate. Therefore, limits give us a tool to analyze and compare how certain functions behave in extreme conditions, especially towards infinity.

Dominant Terms

Dominant terms play a crucial role when considering the growth rate of functions. As \(x\) approaches infinity, the largest powers of \(x\) in any polynomial or function expression will have the most significant impact on its value.

In \(\sqrt{x^{4}+x}\), the term \(x^4\) vastly overshadows \(x\) when \(x\) is large, hence \(x^4\) is dominant. Similarly, in \(\sqrt{x^{4}-x^{3}}\), \(x^4\) also dominates \(-x^3\) for large \(x\). When simplifying these expressions, the non-dominant terms become negligible. This leads us to approximate both functions with \(\sqrt{x^4} = x^2\) as \(x\) grows very large. Focusing on these dominant terms allows us to simplify complex problems and make accurate comparisons on growth rates for large values of \(x\).

Function Comparison

Function comparison is a method used to determine how two or more functions behave in relation to one another, which is crucial when investigating growth rates.

To compare \(\sqrt{x^{4}+x}\) and \(\sqrt{x^{4}-x^{3}}\), we assess their simplified forms at the limit \(x \to \infty\). Both simplify to \(x^2\) by acknowledging the dominant terms, helping us see that they indeed grow at the same rate as \(x^2\) itself.

By analyzing and equating their dominant terms, we're able to use certain mathematical tools like L'Hôpital's rule or direct simplifications to conclude that these functions grow similarly as \(x\) becomes very large. This type of comparison is especially useful in calculus to determine equivalence in growth, predict long-term behavior, and understand how different functions might converge or deviate as their input skyrockets.