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?
c. √(1+x^4)

검증된 단계별 안내

Recall that the function \(e^x\) grows exponentially as \(x \to \infty\), which means it increases faster than any polynomial or root function.

Analyze the given function \(\sqrt{1 + x^4}\). For large \(x\), the term \(x^4\) dominates inside the square root, so \(\sqrt{1 + x^4} \approx \sqrt{x^4} = x^2\).

Since \(x^2\) is a polynomial function, it grows slower than the exponential function \(e^x\) as \(x \to \infty\).

Therefore, \(\sqrt{1 + x^4}\) grows slower than \(e^x\) as \(x \to \infty\).

To summarize, \(\sqrt{1 + x^4}\) does not grow faster than or at the same rate as \(e^x\); it grows slower.

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주요 개념

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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 any polynomial function as x→∞.

Exponential vs. Polynomial Growth

Exponential functions (e.g., e^x) increase much faster than polynomial functions (e.g., x^n) as x becomes very large. Polynomials grow at a rate proportional to a power of x, while exponentials grow proportionally to a constant raised to the power of x, leading to faster growth.

Asymptotic Comparison Using Limits

To compare growth rates, we use limits such as lim(x→∞) f(x)/g(x). If the limit is zero, f grows slower than g; if infinite, f grows faster; if finite and nonzero, they grow at the same rate. This method helps classify functions like √(1+x^4) relative to e^x.

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?c. √(1+x^4)