4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
g. (1.1)^x

검증된 단계별 안내

Recall that to compare the growth rates of functions as \(x \to \infty\), we analyze their behavior by considering limits or known growth hierarchies: polynomial, exponential, logarithmic, etc.

The function given is \(g(x) = (1.1)^x\), which is an exponential function with base greater than 1.

Since \(x^2\) is a polynomial function and \((1.1)^x\) is an exponential function, exponential functions grow faster than any polynomial function as \(x \to \infty\).

Therefore, \(g(x) = (1.1)^x\) grows faster than \(x^2\) as \(x \to \infty\).

To summarize: \((1.1)^x\) grows faster than \(x^2\), so it does not grow at the same rate or slower.

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Growth Rates of Functions

Growth rates describe how functions behave as the input becomes very large. Comparing growth rates helps determine which functions increase faster, slower, or at the same pace. For example, polynomial functions like x² grow slower than exponential functions like (1.1)^x as x approaches infinity.

Polynomial vs. Exponential Functions

Polynomial functions are expressions involving powers of x, such as x², and grow at a rate proportional to a power of x. Exponential functions, like (1.1)^x, grow by repeatedly multiplying by a constant base, leading to much faster growth than any polynomial as x becomes large.

Limits and Asymptotic Behavior

Limits describe the behavior of functions as the input approaches infinity. Analyzing limits helps compare growth by examining the ratio of two functions as x→∞. If the limit of their ratio is zero, one grows slower; if infinite, it grows faster; if finite and nonzero, they grow at the same rate.

4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?g. (1.1)^x