Order of convergence of iterative methods

Using the method of tangents, construct a second-order iterative process for computing $\sqrt{a}$.

Prove that Newton's method (the method of tangents) has second order.

Prove that the iterative formula $x_{n+1} = \frac{x_{n}^{3}+2 a x_{n}}{3 x_{n}^{2}+a}$ for solving the equation $x^{2} = a, a > 0$, has third order.

Using the expansion of the function by Taylor's formula up to second order, obtain an iterative formula for solving the equation $f(x) = 0$ and prove that it has third order: