Question

consider the statement if $0\le f\left(x\right)\le g\left(x\right)$for every x>0 and$\underset{x\to \infty }{\lim }\frac{f\left(x\right)}{g\left(x\right)}=3$, then the improper integrals ${\int }_0^{\infty }g\left(x\right)dxand{\int }_0^{\infty }f\left(x\right)dx$both converge. The objective is to determine whether statement is true or false.

Short answer

False

Step-by-step solution

To determine the statement converge or not

Consider the function $g\left(x\right)=\frac{1}{x^2}andf\left(x\right)=\frac{3}{x^2}$.

The value of $\underset{x\to \infty }{\lim }\frac{f\left(x\right)}{g\left(x\right)}$is:

$\underset{x\to \infty }{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\underset{x\to \infty }{\lim }3$

= 3

Therefore, both the integrals, ${\int }_0^{\infty }g\left(x\right)dx={\int }_0^{\infty }\frac{1}{x^2}dx$and${\int }_0^{\infty }f\left(x\right)dx={\int }_0^{\infty }\frac{3}{x^2}$diverges.

Therefore the given statement is false.