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?
a. x-3

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Recall that the function \(e^x\) is an exponential function, which grows faster than any polynomial function as \(x \to \infty\).

Compare the given function \(x - 3\) to \(e^x\): since \(x - 3\) is a linear polynomial, it grows much slower than \(e^x\) as \(x\) becomes very large.

To determine growth rates, consider the limit \(\lim_{x \to \infty} \frac{f(x)}{e^x}\) for the function \(f(x) = x - 3\).

Evaluate the limit \(\lim_{x \to \infty} \frac{x - 3}{e^x}\). If this limit is 0, then \(x - 3\) grows slower than \(e^x\).

Since the limit tends to 0, conclude that \(x - 3\) grows slower than \(e^x\) as \(x \to \infty\).

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Key Concepts

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

Growth Rates of Functions

Growth rates describe how functions behave as the input approaches infinity. Comparing growth rates helps determine which functions increase faster, slower, or at the same pace. For example, polynomial functions grow slower than exponential functions like e^x as x→∞.

Exponential Functions

Exponential functions have the form a^x, where the variable is in the exponent. The function e^x grows rapidly and dominates polynomial and logarithmic functions as x approaches infinity. Understanding e^x is key to comparing growth rates.

Limits and Asymptotic Behavior

Limits describe the behavior of functions as the input approaches a particular value, often infinity. Asymptotic behavior focuses on how functions compare in growth by examining the limit of their ratio. This helps classify functions as growing faster, slower, or at the same rate.

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