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?
a. 10x^4 + 30x + 1

검증된 단계별 안내

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

Identify the given function: \(10x^4 + 30x + 1\), which is a polynomial of degree 4.

Compare the growth rates by considering the limit \(\lim_{x \to \infty} \frac{10x^4 + 30x + 1}{e^x}\).

Since \(e^x\) grows faster than any polynomial, this limit approaches 0, indicating that \(10x^4 + 30x + 1\) grows slower than \(e^x\) as \(x \to \infty\).

Therefore, the function \(10x^4 + 30x + 1\) grows slower than \(e^x\) as \(x\) approaches infinity.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 as others, especially as x approaches infinity.

Exponential Functions

Exponential functions like e^x grow by continuously multiplying by a constant base raised to the variable power. They increase faster than any polynomial function as x approaches infinity, making them a benchmark for comparing growth rates.

Polynomial Functions

Polynomial functions are sums of terms with variables raised to whole number powers, such as 10x^4 + 30x + 1. Their growth rate is dominated by the highest power term and is slower than exponential functions as x approaches infinity.

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?a. 10x^4 + 30x + 1