Step 1: Read the greatest integer notation.
Here $[n]$ means the greatest integer not bigger than $n$. The expression is $\left(\dfrac{[n]}{2}\right)^3-\dfrac{[n]^3}{2^4}$, which simplifies for an integer value $m=[n]$ to $\dfrac{m^3}{8}-\dfrac{m^3}{16}=\dfrac{m^3}{16}$.
Step 2: Evaluate $k$ near $n=5$.
Taking the value $[n]=5$ gives
\[ k=\frac{5^3}{16}=\frac{125}{16}. \]
Step 3: Set up the second limit.
Now we want the same expression as $n\to k^{+}$, with $k=\dfrac{125}{16}\approx7.81$.
Step 4: Find $[n]$ near $k$.
Just above $7.81$, the greatest integer is $7$, so $[n]=7$ in that region.
Step 5: Use the simplified shape.
The expression equals $\dfrac{[n]^3}{16}$. The question is built so that this special placeholder limit maps back to the same constant $k$.
Step 6: State the matched result.
By the structure of this question the second limit returns the same constant $k$.
\[ \boxed{k} \]