Q. 65 - 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. 

$u\left(x\right) = x+100, v\left(x\right) = x$

Step-by-Step Solution

Answer

$\underset{x\to \infty }{\lim }\frac{v\left(x\right)}{u\left(x\right)}=\underset{x\to \infty }{\lim }\frac{u\left(x\right)}{v\left(x\right)}=1$

Step 1. Given information

$u\left(x\right) = x+100, v\left(x\right) = x$

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

$\begin{array}{r}\underset{x\to \mathrm{\infty }}{lim} \frac{u\left(x\right)}{v\left(x\right)}=\underset{x\to \mathrm{\infty }}{lim} \frac{x+100}{x}\\ =\underset{x\to \mathrm{\infty }}{lim} \left(1+\frac{100}{x}\right)\\ =1+0\\ =1\end{array}$

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

$\begin{array}{r}\underset{x\to \mathrm{\infty }}{lim} \frac{v\left(x\right)}{u\left(x\right)}=\underset{x\to \mathrm{\infty }}{lim} \frac{x}{x+100}\\ =\underset{x\to \mathrm{\infty }}{lim} \frac{x+100-100}{x+100}\\ =\underset{x\to \mathrm{\infty }}{lim} \left(1-\frac{100}{x+100}\right)\\ =1\end{array}$