Question

A function f dominates another function g as $x\to \infty$if $f\left(x\right),g\left(x\right)$both grow without bound as $x\to \infty$and if data-custom-editor="chemistry" $\underset{x\to \infty }{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\infty$.

Intuitively, f dominates gas $x\to \infty$if $f\left(x\right)$is very much larger than $g\left(x\right)$for very large values of x. Use limits to determine whether $u\left(x\right)$dominates data-custom-editor="chemistry" $\mathrm{v}\left(\mathrm{x}\right)$or $v\left(x\right)$dominates $u\left(x\right)$or neither.

$$\begin{gathered} u\left(x\right)=x^{10}+3, \\ v\left(x\right)={10}^x+3 \end{gathered}$$

Short answer

The function $u\left(x\right)$dominates $v\left(x\right)$so localid="1648749438712" $\underset{x\to \infty }{\lim }\frac{x^{10}+3}{{10}^x+3}=\infty$.

Step-by-step solution

Step 1. Given information.

Consider the given question,

$$\begin{gathered} u\left(x\right)=x^{10}+3, \\ v\left(x\right)={10}^x+3 \end{gathered}$$

Step 2. Use the limits.

Using the limits,

$\underset{x\to \infty }{\lim }\frac{u\left(x\right)}{v\left(x\right)}=\underset{x\to \infty }{\lim }\frac{x^{10}+3}{{10}^x+3}$

As$x^{10}>{10}^x$,

localid="1648749428944" $\underset{x\to \infty }{\lim }\frac{x^{10}+3}{{10}^x+3}=\infty$

Again, localid="1648749419242" $\underset{x\to \infty }{\lim }\frac{v\left(x\right)}{u\left(x\right)}=\underset{x\to \infty }{\lim }\frac{{10}^x+3}{x^{10}+3}$

As $x^{10}>{10}^x$,

localid="1648749422931" $\underset{x\to \infty }{\lim }\frac{{10}^x+3}{x^{10}+3}=0$