These are my thoughts on Newton's Method after watching the attached videos. I am Looking for feedback on my summation. Am I correct in my assessment or no?

https://www.youtube.com/watch?v=1uN8cBGVpfs

https://www.youtube.com/watch?v=-DpZOZTsdvg

https://www.youtube.com/watch?v=xdLgTDlFwrc

https://www.youtube.com/watch?v=j6ikEASjbWE

Can Newton's Approximation Method ever fail?

Newton's method fails in instances where the derivative is zero. This occurs if the tangent line drawn is horizontal and therefore never hits the x-axis failing to locate a root. A point that is a zero derivative in the denominator portion of the equation will cause it to become undefined.

A) Please comment on what you have learned from these two videos.

B) Include the TWO requirements for when Newton Approximation will work.

The videos show Newton's method using a tangent line in an equation where it could help solve what the values of x could possibly be. To get the "x-intercept", it could be the middle point of the tangent lines that can be drawn by the function of x or f(x). In order to obtain them accurately, there are correlations on the values that you get from xn+1.

The two requirements necessary for Newton's Approximation work include that a good guess is given for the f(x) equation so that the Method can be applied. Next, the approximation should be started initially with a function that produces continuous derivatives.