Chapter 16.1, Problem 55E

To determine

(a)

The mathematical proof of the statement stating that using L'Hopital Rule, $f(x)=\frac{e^{-x}-e^{-ax}}{x}$, though not defined at x = 0, can be made continuous by assigning the value $f(0)=a-1$

To determine

(b)

The mathematical proof of the statement stating that $|f(x)|\le e^{-x}+e^{-ax}$ for $x>1$ using triangle inequality and the mathematical proof of the statement stating that I(a) converges by applying the Comparison Theorem

To determine

(c)

The mathematical proof of the equation $I(a)={\displaystyle \underset{0}{\overset{\infty }{\int }}{\displaystyle \underset{1}{\overset{a}{\int }}{e}^{-xy}dydx}}$

To determine

(d)

By interchanging the order of integration, the mathematical proof of the equation

$I(a)=\ln a-\underset{T\to \infty }{\lim }{\displaystyle \underset{1}{\overset{a}{\int }}\frac{e^{-Ty}}{y}dy}$

To determine

(e)

The mathematical proof of the statement stating the limit in $I(a)=\ln a-\underset{T\to \infty }{\lim }{\displaystyle \underset{1}{\overset{a}{\int }}\frac{e^{-Ty}}{y}dy}$ is zero by using the Comparison Theorem.