Problem 10 - Calculus Early Transcendentals

Question

In terms of limits, what does it mean for the rates of growth of \(f\) and \(g\) to be comparable as \(x \rightarrow \infty ?\)

Step-by-Step Solution

Answer

Answer: The expression representing comparable rates of growth as x approaches infinity is: $$\lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = C$$ where C is a finite and nonzero constant.

Step 1: Understand the Rates of Growth

The rates of growth of functions \(f(x)\) and \(g(x)\) refer to how fast these functions grow as the input variable \(x\) increases. If two functions have comparable rates of growth, it means that their growth rates are roughly similar, i.e., one does not dominate the other significantly as \(x\) approaches infinity.

Step 2: Define Comparable Growth Rates

We can define the comparable rates of growth using the ratio of the functions as \(x\) approaches infinity. If the limit of the ratio of the functions as \(x\) approaches infinity is finite and nonzero, we say that the rates of growth of \(f(x)\) and \(g(x)\) are comparable as \(x \rightarrow \infty\).

Step 3: Write the Expression for Comparable Rates of Growth using Limit Notation

In terms of limits, we can write the definition of comparable growth rates as: $$\lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = C$$ where \(C\) is a constant that is finite and nonzero. If this condition holds, then it means that the rates of growth of \(f(x)\) and \(g(x)\) are comparable as \(x\) approaches infinity.