Determine the smallest wavelength of radiation that can be emitted...

The smallest wavelength of emitted radiation corresponds to the maximum energy transition, which occurs when an electron transitions from $n = \infty$ to $n = 1$.

Using the Rydberg formula for hydrogen-like species:

$\dfrac{1}{\lambda} = R Z^2 \left( \dfrac{1}{n_1^2} - \dfrac{1}{n_2^2} \right)<br><br>For \text{Li}^{++}, the atomic number$Z = 3$. Substituting$n_1 = 1$and$n_2 = $\infty:<br><br>\dfrac{1}{\lambda} =$R (3)^2 \left( \dfrac{1}{1^2} - 0 \right)$<br><br>$\dfrac{1}{\lambda} = 9R$<br><br>$\lambda = \dfrac{1}{9R}$<br><br>Since$\dfrac{1}{R} \approx 91.2 \text{ nm}$:<br><br>$\lambda = \dfrac{91.2 \text{ nm}}{9} \approx 10.13 \text{ nm}$<br><br>The closest value among the given options is$10 \text{ nm}$.$