Answer:
1
Step-by-step explanation:
We are asked to find the value of [tex]x[/tex] such that:
[tex]x=\frac{1}{2-\frac{1}{2-\cdots}}[/tex]
Notice that we can rewrite this as:
[tex]x=\frac{1}{2-x}[/tex]
[tex]\frac{x}{1}=\frac{1}{2-x}[/tex]
Cross multiply:
[tex]x(2-x)=1[/tex]
Expand:
[tex]2x-x^2=1[/tex]
Subtract 1 on both sides:
[tex]2x-x^2-1=0[/tex]
Reorder:
[tex]-x^2+2x-1=0[/tex]
Multiply both sides by -1:
[tex]x^2-2x+1=0[/tex]
Factor as a binomial square on left:
[tex](x-1)^2=0[/tex]
This implies [tex]x-1=0[/tex].
Add 1 on both sides:
[tex]x=1[/tex]
