The Newton-Raphson method is a powerful numerical technique used to find the roots of equations. This iterative method begins with an initial guess for the root and improves upon that guess using the derivative of the function. The underlying principle is that if we have a sufficiently accurate approximation of the root, we can use the tangent line at that point to derive a better approximation.

The formula for the Newton-Raphson method is given by:

In this expression, $x_n$ represents the current approximation to the root, $f(x_n)$ denotes the value of the function at $x_n$, and $f'(x_n)$ is the derivative of the function evaluated at $x_n$.

To apply the method, we start with an initial guess, denoted as $x_0$. We then iteratively use the formula to compute subsequent approximations: $x_1$, $x_2$, $x_3$, and so forth, until we achieve the desired level of accuracy. The Newton-Raphson method can converge rapidly to the root, provided that the initial guess is sufficiently close to the actual root and the function behaves well in the vicinity of that root.

Despite its strengths, the Newton-Raphson method has certain limitations. It may fail to converge if the initial guess is too distant from the root, or if the function exhibits a flat region or a sharp turn near the root. Additionally, if multiple roots or extrema exist, the method may converge to an unintended root or to a local minimum or maximum instead. Therefore, it is essential to carefully assess both the results and the conditions necessary for convergence when using this method.