Suppose you attempt to use Newton's method to approximate a solution of the equation

x$312$x$+7=03-12+7=0.$

Letx$0=20=2$be the initial approximation, and then calculatex$11$andx$22.$ If the value is undefined, enterDNE.

x$1=1=$

x$2=2=$

What do you notice?

Newton's method fails because f$\text{'}($x$0)=0(0)=0$ but f$($x$0)0(0)0.$

Newton's method fails because the approximations are getting larger in absolute value.

Newton's method fails because the approximations are alternating between two numbers.

Newton's method is successful at approaching a number so far.