= Solution
For fixed real $t$, <Taylor's theorem> for the <exponential function> gives
$$
e^{t/\sqrt{k}}=1+\frac t{\sqrt{k}}+\frac{t^2}{2k}+O(k^{-3/2}).
$$
Substituting into the previous <moment-generating function>,
$$
\log M_{Y_k}(t)=\frac{t^2}{2}+O(k^{-1/2}),\qquad
M_{Y_k}(t)\longrightarrow e^{t^2/2}.
$$
The <continuity theorem for moment-generating functions> states that, if the <moment-generating functions> of $Z_n$ and $Z$ are finite on a common open <interval> containing zero and $M_{Z_n}(t)\to M_Z(t)$ throughout that interval, then $Z_n\xrightarrow{\mathrm d}Z$. Here finiteness holds for all real $t$, and the limit is the <moment-generating function of a standard normal variable>. Therefore
$$
\boxed{Y_k\xrightarrow{\mathrm d}N(0,1).}
$$
This proves the required instance of the <Poisson central limit theorem> directly from <moment-generating functions>; no appeal to a general <central limit theorem> is needed.
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