2. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?
g. e^(cos(x))

Verified step by step guidance

1. Recall that the growth rate of a function as \(x \to \infty\) is determined by its dominant behavior for large \(x\). The function \(e^x\) grows exponentially and very rapidly as \(x\) increases.
2. Analyze the given function \(g(x) = e^{\cos(x)}\). Since \(\cos(x)\) oscillates between \(-1\) and \(1\), the exponent \(\cos(x)\) does not grow without bound; it remains bounded.
3. Because \(\cos(x)\) is bounded, \(e^{\cos(x)}\) oscillates between \(e^{-1}\) and \(e^{1}\), which are constant values. This means \(g(x)\) remains bounded and does not increase without limit as \(x \to \infty\).
4. Compare this behavior to \(e^x\), which grows without bound. Since \(g(x)\) remains bounded, it grows much slower than \(e^x\) as \(x \to \infty\).
5. Therefore, \(g(x) = e^{\cos(x)}\) grows slower than \(e^x\) as \(x \to \infty\).

Key Concepts

Exponential Growth and Limits

Exponential growth describes functions like e^x that increase rapidly as x approaches infinity. Understanding limits helps compare how fast functions grow by analyzing their behavior as x→∞, determining if one function outpaces, matches, or lags behind another.

Behavior of Composite Functions

Composite functions like e^(cos(x)) combine an exponential with a bounded function (cos(x)). Since cos(x) oscillates between -1 and 1, e^(cos(x)) oscillates between e^(-1) and e^1, remaining bounded and not growing without limit as x→∞.

Growth Rate Comparison

Comparing growth rates involves analyzing dominant terms and their limits. A function that remains bounded or oscillates does not grow faster than an unbounded exponential like e^x. Functions with the same dominant exponential term grow at the same rate, while bounded functions grow slower.