Question

what is the limit of comparison test for improper integrals?

Short answer

if f and g are continuous ,positive functions for all values of x and .

then 1) if $0<k<\infty$then ${\int }_a^{\infty }g\left(x\right)dx$converges then ${\int }_a^{\infty }f\left(x\right)dx$converges.

2) if k=0 then ${\int }_a^{\infty }g\left(x\right)dx$converges then${\int }_a^{\infty }f\left(x\right)dx$converges.

3) if k=$\infty$then ${\int }_a^{\infty }g\left(x\right)dx$converge then${\int }_a^{\infty }f\left(x\right)dx$converges.

Step-by-step solution

the limit of comparison test for improper integrals

if f and g are continuous ,positive functions for all values of x and $\underset{x\to \infty }{\lim }\frac{f\left(x\right)}{g\left(x\right)}=k$.

then 1) if localid="1651481834930" $0<k<\infty$then localid="1651481820957" ${\int }_a^{\infty }g\left(x\right)dx$converges thenlocalid="1651481826895" ${\int }_a^{\infty }f\left(x\right)dx$converges.

2) if k=0 then ${\int }_a^{\infty }g\left(x\right)dx$converges then${\int }_a^{\infty }f\left(x\right)dx$converges.

3) if k=$\infty$then${\int }_a^{\infty }g\left(x\right)dx$converge then${\int }_a^{\infty }f\left(x\right)dx$converges.