(i) Show that the logarithmic utility function \(u(x)=\log (a+x)\) is the limiting case of a power utility function \(u(x)=\frac{1}{b-1}\left((a+b x)^{(b-1) / b}-1\right)\) for \(b \rightarrow 1\).

(ii) Show that the exponential utility function \(u(x)=-\exp (-x / a)\) is the limiting case of a power utility function \(u(x)=\frac{1}{b-1}\left(1+\frac{b x}{a}\right)^{\frac{b-1}{b}}\) for \(b \rightarrow 0\).

(iii) Show that the utility function \(u(x)=\frac{1-\mathrm{e}^{a x}}{a}\) tends to a linear utility function as \(a \rightarrow 0\).