Which of the following is NOT a possible value of

Hello Gmat_mission.

Let's take a look at your question.

We have the fraction $$\frac{1}{4-x}.$$ We know that a fraction is equal to 0 if, and only if, the numerator is equal to 0. Since 1 is different from 0, we have that the given fraction can't be equal to 0.

This is why the correct answer is the option C.

Now, you asked how can we obtain 4/17 and 17/4, this can be known by solving the equations $$\frac{1}{4-x}=\frac{4}{17}\ \ \ \Rightarrow\ \ 4-x=\frac{17}{4}\ \Rightarrow\ \ x=4-\frac{17}{4}=-\frac{1}{4}.$$ And $$\frac{1}{4-x}=\frac{17}{4}\ \ \ \Rightarrow\ \ 4-x=\frac{4}{17}\ \Rightarrow\ \ x=4-\frac{4}{17}=\frac{64}{17}.$$ I hope it can help you.