For each specified pair of functions \(G(t)\) and \(g(t)\), determine the value of \(\alpha\) and \(c\) so that \(G(t) \asymp c t^{\alpha-1}\) as \(t ightarrow \infty\) and determine if there is a function \(g(t) \asymp d t^{\alpha}\) for some \(d\) as \(t ightarrow \infty\) where \(c\) and \(d\) are real constants. State whether Theorem 1.17 is applicable in each case.

a. \(G(t)=2 t^{4}+t\)

b. \(G(t)=t+t^{-1}\)

c. \(G(t)=t^{2}+\cos (t)\)

d. \(G(t)=t^{1 / 2}+\cos (t)\)