Newton's Method will converge to a true solution if you have a good initial approximation. If you don't it may not converge at all. Consider, for example, the equation $$f(x) = x(x-1)(x+1)=x^3-x = 0.$$ Obviously, the solutions are $$x=\quad -1,\quad 0, \quad 1.$$ If we start Newton's Method with $x_0$ being close to one of these solutions we will get convergence to that solution. On the other hand, note that $$f'(x) = 3x^2-1$$ is zero when $$x=\pm\frac{1}{\sqrt{3}}.$$ Thus the tangent will be horizontal in those two cases, and Newton's can't even be carried out.
In this problem we'll investigate what happens in the contrived case that $$x_0= \frac{\sqrt{5}}{5}.$$ You can enter $x_0$ into WeBWorK as $\tt{sqrt}(5)/5$. Try it:
$x_0 =$ ,
Now do your computations using exact arithmetic, and you'll recognize a pattern:
$x_1 =$ ,
$x_2 =$ ,
$x_3 =$ ,
$x_4 =$ ,
Draw a picture to see what's going on. Note, however, that Newton's method may fail in many different ways. A detailed analysis of Newton's method and related methods is a huge subject and well beyond the scope of our class.