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

Verified step by step guidance

1. Recall that the function \(e^x\) is an exponential function, which generally grows faster than any polynomial function as \(x \to \infty\).
2. Identify the given function: \(10x^4 + 30x + 1\), which is a polynomial of degree 4.
3. Compare the growth rates by considering the limit \(\lim_{x \to \infty} \frac{10x^4 + 30x + 1}{e^x}\).
4. Since \(e^x\) grows faster than any polynomial, this limit approaches 0, indicating that \(10x^4 + 30x + 1\) grows slower than \(e^x\) as \(x \to \infty\).
5. Therefore, the function \(10x^4 + 30x + 1\) grows slower than \(e^x\) as \(x\) approaches infinity.

Key Concepts

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.