Q. 67 - Calculus

Question

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.

$u\left(x\right) = 100x^2, v\left(x\right) = 2x^{100}$

Step-by-Step Solution

Answer

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

Step 1. Given information

$u\left(x\right) = 100x^2, v\left(x\right) = 2x^{100}$

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

$$\begin{gathered} \underset{x\to \infty }{\lim }\frac{u\left(x\right)}{v\left(x\right)}=\underset{x\to \infty }{\lim }\frac{100x^2}{2x^{100}} \\ =50 \underset{x\to \infty }{\lim }\frac{1}{x^{100-2}} \\ =50 \underset{x\to \infty }{\lim }\frac{2}{x^{98}} \\ =0 \end{gathered}$$

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

$$\begin{gathered} \underset{x\to \infty }{\lim }\frac{v\left(x\right)}{u\left(x\right)}=\underset{x\to \infty }{\lim }\frac{2x^{100}}{100x^2} \\ =\frac{1}{50} \underset{x\to \infty }{\lim }{x}^{100-2} \\ =50 \underset{x\to \infty }{\lim }{x}^{98} \\ =\infty \end{gathered}$$