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

Verified step by step guidance

1. Recall that the function \(e^x\) is an exponential function with base \(e\), and its growth rate as \(x \to \infty\) is very fast compared to polynomial or logarithmic functions.
2. Analyze the given function \(f(x) = \frac{e^x}{2}\). This can be rewritten as \(f(x) = \frac{1}{2} e^x\).
3. Since \(f(x)\) is just \(e^x\) multiplied by a constant factor \(\frac{1}{2}\), the growth rate of \(f(x)\) as \(x \to \infty\) is the same as that of \(e^x\).
4. In general, multiplying an exponential function by a positive constant does not change its growth rate classification; it only scales the function vertically.
5. Therefore, \(f(x) = \frac{e^x}{2}\) grows at the same rate as \(e^x\) as \(x \to \infty\).

Key Concepts

Exponential Growth

Exponential growth describes functions where the variable appears in the exponent, such as e^x. These functions increase rapidly as x approaches infinity, outpacing polynomial and logarithmic functions. Understanding the nature of e^x is essential for comparing growth rates.

Growth Rate Comparison Using Limits

To compare growth rates of functions as x→∞, we use limits of their ratios. If the limit of f(x)/g(x) is zero, f grows slower; if it is a finite nonzero constant, they grow at the same rate; if infinite, f grows faster. This method helps classify functions by their relative growth.

Constant Multipliers and Their Effect on Growth

Multiplying a function by a constant, like (e^x)/2, does not change its growth rate class. Constants scale the function's value but do not affect how fast it grows compared to other functions as x→∞. Recognizing this helps in identifying equivalent growth rates.