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?
c. √x

검증된 단계별 안내

Recall the growth rates of common functions as \(x \to \infty\). The exponential function \(e^x\) grows faster than any polynomial or root function.

Identify the function given: \(\sqrt{x}\), which is equivalent to \(x^{1/2}\), a root function and a type of power function.

Compare the growth of \(\sqrt{x}\) to \(e^x\). Since \(e^x\) grows exponentially and \(\sqrt{x}\) grows polynomially, \(e^x\) grows faster than \(\sqrt{x}\) as \(x \to \infty\).

Conclude that \(\sqrt{x}\) grows slower than \(e^x\) as \(x \to \infty\).

Summarize: \(\sqrt{x}\) does not grow faster than or at the same rate as \(e^x\); it grows slower.

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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, exponential functions like e^x grow faster than polynomial or root functions as x→∞.

Exponential Functions

Exponential functions have the form a^x, where the variable is in the exponent. The function e^x is a fundamental exponential function that grows rapidly as x increases. Its growth rate surpasses that of any polynomial or root function for large x.

Polynomial and Root Functions

Polynomial functions involve variables raised to constant powers, while root functions are fractional powers (e.g., √x = x^(1/2)). These functions grow slower than exponential functions like e^x as x→∞, meaning their values increase but at a much slower rate.

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?c. √x