= Solution
Use the <characteristic function> to establish <convergence in distribution> by the <Lévy continuity theorem>, especially when exponential moments do not exist; a <Cauchy distribution>, for example, has a <characteristic function> but no finite <moment-generating function> on a neighborhood of zero.
Use the <moment-generating function> for an exponential-tail estimate: for $t>0$ with $M_X(t)<\infty$, <Markov inequality> gives
$$
P(X\geq x)\leq e^{-tx}M_X(t).
$$
Optimizing over $t$ produces a <Chernoff bound>. The modulus-one <characteristic function> does not directly give this positive exponential bound.
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