Question

1. x^{100} \qquad 20. (e^x + e^{-x})/2 \qquad 0, 100

in exercises 21–24, determine whether the function grows faster than x^2, at the same rate as x^2, or slower than x^2 as x\to\infty.

1. x^3 + 3 \qquad 22. \qquad 24. \qquad 37.

Explanation:

Step1: Recall the definition of growth rate

To determine the growth rate of a function \( f(x) \) relative to \( g(x) \) as \( x \to \infty \), we use the limit \( \lim_{x \to \infty} \frac{f(x)}{g(x)} \). If the limit is \( \infty \), \( f(x) \) grows faster; if \( 0 \), slower; if a finite non - zero number, same rate. Here \( f(x)=x^{3}+3 \), \( g(x)=x^{2} \).

Step2: Compute the limit

Calculate \( \lim_{x \to \infty} \frac{x^{3}+3}{x^{2}}=\lim_{x \to \infty}(x+\frac{3}{x^{2}}) \). As \( x \to \infty \), \( x\to\infty \) and \( \frac{3}{x^{2}}\to0 \), so the limit is \( \infty \).

Answer:

The function \( x^{3}+3 \) grows faster than \( x^{2} \) as \( x\to\infty \).