Chapter 7.1, Problem 67E

(a)

To determine

To prove: That $\frac{1}{x+1}<{\displaystyle \underset{x}{\overset{x+1}{\int }}\frac{1}{t}dt}<\frac{1}{x}$ for $x>0$.

(b)

To determine

To prove: That $\frac{1}{x+1}<\ln \left(1+\frac{1}{x}\right)<\frac{1}{x}$ from part (a).

(c)

To determine

To prove: That $e^{\frac{x}{x+1}}<{\left(1+\frac{1}{x}\right)}^x<e$ from part (b).

(d)

To determine

To prove: That $\underset{x\to \infty }{\lim }{\left(1+\frac{1}{x}\right)}^x=e$ from part (c).