Chapter 5.10, Problem 48ES

(a) Give a geometric argument to show that

$\frac{1}{x+1}<{\displaystyle {\int }_x^{x+1}\frac{1}{t}dt<\frac{1}{x},}x>0$

(b) Use the result in part (a) to prove that

$\frac{1}{x+1}<\ln \left(1+\frac{1}{x}\right)\text{<}\frac{1}{x}\text{, }x>0$

(c) Use the result in part (b) to prove that

$e^{x/\left(x+1\right)}<{\left(1+\frac{1}{x}\right)}^x<e,x>0$

and hence that

$\underset{x\to +\infty }{\lim }{\left(1+\frac{1}{x}\right)}^x=e$

(d) Use the result in part (b) to prove that

${\left(1+\frac{1}{x}\right)}^x<e<{\left(1+\frac{1}{x}\right)}^{x+1},x>0$