IMathAS-libretexts

Consider the Type I improper integral

\displaystyle {\int_{{0}}^{\infty}}{2}{e}^{{-{x}}}{\left.{d}{x}\right.}

.

To determine whether the integral is divergent or convergent, do the following:

a) Find the value of

\displaystyle {\int_{{0}}^{{t}}}{2}{e}^{{-{x}}}{\left.{d}{x}\right.}

. Note that your answer will be in terms of $\displaystyle {t}$ .

\displaystyle {\int_{{0}}^{{t}}}{2}{e}^{{-{x}}}{\left.{d}{x}\right.}=

b) Now find the limit of your answer as

\displaystyle {t}

approaches infinity. If the limit exists enter the limit value. If the limit does not exist, enter DNE.

\displaystyle \lim_{{{t}\rightarrow\infty}}{\int_{{0}}^{{t}}}{2}{e}^{{-{x}}}{\left.{d}{x}\right.}=

c) Which of the following is true?

The integral $\displaystyle {\int_{{0}}^{\infty}}{2}{e}^{{-{x}}}{\left.{d}{x}\right.}$ is divergent.
The integral $\displaystyle {\int_{{0}}^{\infty}}{2}{e}^{{-{x}}}{\left.{d}{x}\right.}$ is neither convergent nor divergent.
The integral $\displaystyle {\int_{{0}}^{\infty}}{2}{e}^{{-{x}}}{\left.{d}{x}\right.}$ is convergent.