A sequence of real numbers $x_{1}, x_{2}, x_{3}, \dots$ satisfies the recurrence relation $x_{n+1} = \dfrac{5x_{n} + 2}{2x_{n} + 3}\quad\text{for all } n \ge 1.$

1. Set up the equation for the convergence limit.

Since the sequence converges to $l$, we can take the limit as $n \to \infty$ on both sides of the recurrence relation, setting $x_{n+1} = l$ and $x_n = l$:
$l = \frac{5l + 2}{2l + 3}$

2. Rearrange the equation into a standard quadratic form.

$l(2l + 3) = 5l + 2$
$2l^2 + 3l = 5l + 2$
$2l^2 - 2l - 2 = 0$
$l^2 - l - 1 = 0$

3. Solve the quadratic equation to find the exact values of $l$.

Using the quadratic formula:
$l = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-1)}}{2(1)}$
$l = \frac{1 \pm \sqrt{1 + 4}}{2}$
$l = \frac{1 \pm \sqrt{5}}{2}$

1. Calculate the first few terms to observe the sequence's behaviour.

$x_1 = 3$
$x_2 = \frac{5(3)+2}{2(3)+3} = \frac{17}{9} \approx 1.889$
$x_3 = \frac{5(\frac{17}{9})+2}{2(\frac{17}{9})+3} = \frac{103}{61} \approx 1.689$
$x_4 = \frac{5(\frac{103}{61})+2}{2(\frac{103}{61})+3} = \frac{637}{389} \approx 1.638$

The values are strictly decreasing.

2. Determine the value the sequence converges to by observing long-term behaviour.

As $n \to \infty$, using the calculator to evaluate subsequent terms shows that $x_n$ approaches approximately $1.61803$, which corresponds to $\frac{1+\sqrt{5}}{2}$ (the positive possible exact value of $l$ found in part (a)).

Therefore, the sequence decreases and converges to $\frac{1+\sqrt{5}}{2}$.

1. Make $x_n$ the subject of the recurrence relation.

$x_{n+1} = \frac{5x_n + 2}{2x_n + 3}$
$x_{n+1}(2x_n + 3) = 5x_n + 2$
$2x_nx_{n+1} + 3x_{n+1} = 5x_n + 2$
$x_n(2x_{n+1} - 5) = 2 - 3x_{n+1}$
$x_n = \frac{2 - 3x_{n+1}}{2x_{n+1} - 5}$

2. Iterate backwards from $x_5$ to find $x_1$ using a graphing calculator.

Using a graphing calculator to generate the previous terms sequentially:
$x_4 = \frac{2 - 3\left(\frac{3503}{2158}\right)}{2\left(\frac{3503}{2158}\right) - 5} = \frac{563}{344}$
$x_3 = \frac{2 - 3\left(\frac{563}{344}\right)}{2\left(\frac{563}{344}\right) - 5} = \frac{91}{54}$
$x_2 = \frac{2 - 3\left(\frac{91}{54}\right)}{2\left(\frac{91}{54}\right) - 5} = \frac{15}{8}$
$x_1 = \frac{2 - 3\left(\frac{15}{8}\right)}{2\left(\frac{15}{8}\right) - 5} = \frac{29}{10}$