Chapter 4.2, Problem 12ES

a.

To determine

To show: The limit $\underset{h\to 0}{\lim }{\left(\frac{2+h}{2-h}\right)}^{\frac{1}{h}}=e$

b.

To determine

The approximations to e using the formula $N\left(h\right)={\left(\frac{2+h}{1-h}\right)}^{\frac{1}{h}}$ for $h=0.04,h=0.02\text{ and }h=0.01$.

c.

To determine

The approximation to e using extrapolation with $h=0.04$.

d.

To determine

To show: The formula $N\left(h\right)=N\left(-h\right)$.

e.

To determine

To show: The formula $e=N\left(h\right)+K_{1}h+K_2{h}^2+K_3{h}^3+K_4{h}^4+\cdots$ reduces to $e=K_2{h}^2+K_4{h}^4+K_6{h}^6+\cdots$.

f.

To determine

An $O\left(h^6\right)$ approximation to e with $h=0.04$ using extrapolation.