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Newton-Raphson follows the tangent line to its x-intercept: xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ). Near a simple root it converges quadratically, roughly doubling the number of correct digits each step.
Newton-Raphson
xₙ₊₁ = xₙ − f(xₙ) / f'(xₙ)
It converges to √2 = 1.4142136 within about four iterations.
At a zero derivative, at an inflection near the root, or from a poor starting guess — then it can cycle or diverge.