109. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
b. f(x)=x, g(x)=x + 1/x

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Identify the given functions: \(f(x) = x\) and \(g(x) = x + \frac{1}{x}\).

As \(x \to \infty\), analyze the behavior of each function. For \(f(x) = x\), the function grows without bound linearly.

For \(g(x) = x + \frac{1}{x}\), note that \(\frac{1}{x} \to 0\) as \(x \to \infty\), so \(g(x)\) behaves like \(x\) plus a very small positive term.

Compare the growth rates by considering the ratio \(\frac{f(x)}{g(x)} = \frac{x}{x + \frac{1}{x}}\). Simplify this expression to understand the limit as \(x \to \infty\).

Evaluate the limit \(\lim_{x \to \infty} \frac{f(x)}{g(x)}\) to determine if it approaches 0, a finite nonzero constant, or infinity, which tells us if \(f\) grows slower, at the same rate, or faster than \(g\) respectively.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Asymptotic Behavior of Functions

Asymptotic behavior describes how functions behave as the input grows very large (x→∞). It helps compare growth rates by focusing on dominant terms and ignoring smaller ones that become insignificant at infinity.

Limits and Limit Comparison

Limits evaluate the value a function approaches as x approaches infinity. Comparing the limit of the ratio f(x)/g(x) as x→∞ determines if one function grows faster, slower, or at the same rate as the other.

Dominant Terms in Functions

The dominant term in a function is the part that grows fastest as x→∞. For example, in g(x) = x + 1/x, the term x dominates because 1/x approaches zero, so the growth rate is mainly determined by x.

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