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 \(\sqrt{x^4+x}\) and \(\sqrt{x^4-x^3}\) grow at the same rate as \(x^2\) as \(x\) approaches infinity.

Step 1: Analyze Dominant Terms

For large values of \(x\), the term \(x^4\) will dominate the expression \(x^4 + x\), so \(\sqrt{x^4 + x} \approx \sqrt{x^4} = x^2\). Similarly, in \(x^4 - x^3\), for large \(x\), the term \(x^4\) will again dominate and \(\sqrt{x^4 - x^3} \approx \sqrt{x^4} = x^2\). This analysis sets the ground for finding the limit and comparing the growth rates with \(x^2\).

Step 2: Evaluate the Limit for \(\sqrt{x^4 + x}\)

To show \(\sqrt{x^4 + x}\) grows at the same rate as \(x^2\), we consider the limit \(\lim_{{x \to \infty}} \frac{\sqrt{x^4 + x}}{x^2}\). Simplifying, we have: \(\lim_{{x \to \infty}} \frac{\sqrt{x^4(1 + \frac{1}{x^3})}}{x^2} = \lim_{{x \to \infty}} \sqrt{1 + \frac{1}{x^3}} = 1\). Thus, \(\sqrt{x^4 + x}\) grows at the same rate as \(x^2\).

Step 3: Evaluate the Limit for \(\sqrt{x^4 - x^3}\)

Similarly, consider the limit \(\lim_{{x \to \infty}} \frac{\sqrt{x^4 - x^3}}{x^2}\). Simplifying, we have: \(\lim_{{x \to \infty}} \frac{\sqrt{x^4(1 - \frac{1}{x})}}{x^2} = \lim_{{x \to \infty}} \sqrt{1 - \frac{1}{x}} = 1\). This shows that \(\sqrt{x^4 - x^3}\) also grows at the same rate as \(x^2\).

Key Concepts

Growth Rate

Understanding growth rate is crucial in asymptotic analysis, as it helps determine how fast a function grows compared to another function when a variable approaches a specific value, like infinity. In our original exercise, we are comparing the growth rates of \( \sqrt{x^4 + x} \) and \( \sqrt{x^4 - x^3} \) to the function \( x^2 \). The goal is to show that as \( x \to \infty \), these functions grow at the same rate as \( x^2 \).

The term "growing at the same rate" means that the quotient of these two functions approaches a non-zero constant. In simpler terms, if two functions grow at the same rate, the graph of their quotient stabilizes to a horizontal line (constant) when the x-values become very large.
When dealing with functions like \( \sqrt{x^4 + x} \) and \( \sqrt{x^4 - x^3} \), observing their dominant terms (discussed later) provides insights. Understanding which part of the expression has the most impact as x becomes very large can help simplify growth rate equations.
When both functions \( \sqrt{x^4 + x} \) and \( \sqrt{x^4 - x^3} \) are divided by \( x^2 \), they simplify to functions behaving closely to a constant (1 in this case), reinforcing that their growth rate matches \( x^2 \).

Limit Evaluation

Limit evaluation is often used to determine how functions behave as they approach infinity. In our scenario, we evaluate limits to compare two functions directly and verify their growth rates to understand how they relate asymptotically to \( x^2 \).

To proceed, we calculate the limit of the ratio of the given functions to \( x^2 \):

• For \( \sqrt{x^4 + x} \), we calculate \( \lim_{{x \to \infty}} \frac{\sqrt{x^4 + x}}{x^2} \).
• The form inside the square root becomes \( \sqrt{x^4(1 + \frac{1}{x^3})} \), simplifying the fraction's limit to \( \lim_{{x \to \infty}} \sqrt{1 + \frac{1}{x^3}} = 1 \).
• This constant result suggests that \( \sqrt{x^4 + x} \) grows proportionally to \( x^2 \) as \( x \to \infty \).
• Similarly, for \( \sqrt{x^4 - x^3} \), we find \( \lim_{{x \to \infty}} \frac{\sqrt{x^4 - x^3}}{x^2} \).
• The form begets \( \sqrt{x^4(1 - \frac{1}{x})} \), which simplifies the limit to \( \lim_{{x \to \infty}} \sqrt{1 - \frac{1}{x}} = 1 \).
• Again, a constant limit of 1 illustrates comparable growth between \( \sqrt{x^4 - x^3} \) and \( x^2 \).

Evaluating these limits confirms our assertion about growth rates being equal.

Dominant Terms

Dominant terms in a function determine its behavior as the variable approaches a particular value, like infinity. It's often the largest degree term in a polynomial expression. Understanding these terms helps simplify complex functions.

Focusing on the exercise, we have:

• In \( x^4 + x \), the dominant term is \( x^4 \). As \( x \) becomes very large, \( x^4 \) grows much more significantly than \( x \), making it the term that predominantly dictates the behavior of the function.
• This allows us to approximate \( \sqrt{x^4 + x} \approx \sqrt{x^4} = x^2 \) when analyzing growth rate.
• In \( x^4 - x^3 \), \( x^4 \) is again the dominant term, as \( x^4 \) dwarfs the \( x^3 \) term for large values of \( x \).
• Accordingly, \( \sqrt{x^4 - x^3} \approx \sqrt{x^4} = x^2 \), reinforcing that \( x^2 \) is the growth baseline for both functions.

By identifying and understanding dominant terms, we streamline the analysis process, confirming that both functions grow at the rate of \( x^2 \). This approach is invaluable not just in these particular cases, but in evaluating growth patterns whenever dealing with similar functional forms.