The largest of the fractions $\dfrac{5}{6}, \dfrac{5}{7}, \dfrac{5}{9}, \dfrac{5}{11}$ is

1. $\dfrac{5}{11}$

2. $\dfrac{5}{9}$

3. $\dfrac{5}{7}$

4. $\dfrac{5}{6}$

Given fractions: $\dfrac{5}{6}, \dfrac{5}{7}, \dfrac{5}{9}, \dfrac{5}{11}$

LCM of the denominators (6, 7, 9, 11) = 1,386

$\Rightarrow \dfrac{5 \times 231}{6 \times 231}, \dfrac{5 \times 198}{7 \times 198}, \dfrac{5 \times 154}{9 \times 154}, \dfrac{5 \times 126}{11 \times 126}\\[1em] \Rightarrow \dfrac{1155}{1386}, \dfrac{990}{1386}, \dfrac{770}{1386}, \dfrac{630}{1386}$

Among fractions with the same denominator, the one with the smaller numerator is smaller.

So, $\dfrac{630}{1386} \lt \dfrac{770}{1386} \lt \dfrac{990}{1386} \lt \dfrac{1155}{1386}$

$\dfrac{5}{11} \lt \dfrac{5}{9} \lt \dfrac{5}{7} \lt \dfrac{5}{6}$

Hence, option 4 is the correct option.