Problem 9 - Calculus Early Transcendentals

Question

In terms of limits, what does it mean for \(f\) to grow faster than \(g\) as \(x \rightarrow \infty ?\)

Step-by-Step Solution

Answer

Answer: For a function \(f(x)\) to grow faster than a function \(g(x)\) as \(x\) approaches infinity, the limit of the ratio \(\frac{f(x)}{g(x)}\) as \(x\) approaches infinity must be infinity, i.e., \(\lim_{x\to\infty} \frac{f(x)}{g(x)} = \infty\).

Step 1: Set up the problem

Given functions \(f(x)\) and \(g(x)\), we want to know what it means for \(f\) to grow faster than \(g\) as \(x \rightarrow \infty\). We will examine the ratio \(\frac{f(x)}{g(x)}\) and find its limit as \(x\) approaches infinity.

Step 2: Determine the growth factor

To determine if \(f\) grows faster than \(g\), we look at the limit of the ratio \(\frac{f(x)}{g(x)}\) as \(x\) approaches infinity. Mathematically, we write this as: $$\lim_{x\to\infty} \frac{f(x)}{g(x)}$$

Step 3: Check the limit

If the limit of the ratio is infinity, that means \(f(x)\) grows faster than \(g(x)\) as \(x\to\infty\). In mathematical terms, we would say: $$\lim_{x\to\infty} \frac{f(x)}{g(x)} = \infty$$ If the limit has other values or does not exist, then \(f\) does not grow faster than \(g\). In conclusion, for function \(f(x)\) to grow faster than function \(g(x)\) as \(x \rightarrow \infty\), their ratio must satisfy the condition: $$\lim_{x\to\infty} \frac{f(x)}{g(x)} = \infty$$