Q. 66 - Calculus

Question

A function f dominates another function g as x → ∞ if f(x) and g(x) both grow without bound as x → ∞ and if Intuitively, f dominates g as x → ∞ if f(x) is very much larger than g(x) for very large values of x. In Exercises 65–74, use limits to determine whether u(x) dominates v(x), or v(x) dominates u(x), or neither. 

$u\left(x\right) = 5x^2+1, v\left(x\right) = x^3$

Step-by-Step Solution

Answer

$\underset{x\to \infty }{\lim }\frac{u\left(x\right)}{v\left(x\right)}=0$ and $\underset{x\to \infty }{\lim }\frac{v\left(x\right)}{y\left(x\right)}=\infty$

Step 1. Given information

$u\left(x\right) = 5x^2+1, v\left(x\right) = x^3$

Step 2. Calculate lim x → ∞ u ( x ) v ( x )

$$\begin{gathered} \underset{x\to \infty }{\lim }\frac{u\left(x\right)}{v\left(x\right)}=\underset{x\to \infty }{\lim }\frac{5x^2+1}{x^3} \\ =0 \end{gathered}$$

Step 3. Calculate lim x → ∞ v ( x ) u ( x )

$$\begin{gathered} \underset{x\to \infty }{\lim }\frac{v\left(x\right)}{u\left(x\right)}=\underset{x\to \infty }{\lim }\frac{x^3}{5x^2+1} \\ =\infty \end{gathered}$$