$$
\(\frac{x-\frac{1}{x}}{1+x}=99\)
$$


And we need to find the value of:

$$
\(\frac{x-\frac{1}{x}}{1-x}\)
$$


The options provided are A. -97, B. -90, C. 0, D. 90, E. 97.
Step 1: Simplify the Given Equation
First, let's simplify the given equation to make it easier to work with. The numerator is $\(x-\frac{1}{x}\)$, which can be combined into a single fraction:

$$
\(x-\frac{1}{x}=\frac{x^2-1}{x}\)
$$


So, the given equation becomes:

$$
\(\frac{\frac{x^2-1}{x}}{1+x}=99\)
$$


This complex fraction can be simplified by multiplying the numerator by the reciprocal of the denominator:

$$
\(\frac{x^2-1}{x} \cdot \frac{1}{1+x}=99\)
$$


Now, notice that $x^2-1$ is a difference of squares, which factors into $(x-1)(x+1)$ :

$$
\(\frac{(x-1)(x+1)}{x} \cdot \frac{1}{x+1}=99\)
$$


The $x+1$ terms cancel out:

$$
\(\frac{x-1}{x}=99\)
$$


Step 2: Solve for $x$
Now, we have a simpler equation:

$$
\(\frac{x-1}{x}=99\)
$$


Let's solve for $x$ :
Multiply both sides by $x$ :

$$
\(x-1=99 x\)
$$


Subtract $x$ from both sides:

$$
\(-1=98 x\)
$$


Divide both sides by 98:

$$
\(x=-\frac{1}{98}\)
$$


Step 3: Compute the Desired Expression
Now that we have $\(x=-\frac{1}{98}\)$, we can compute the expression we're interested in:

$$
\(\frac{x-\frac{1}{x}}{1-x}\)
$$


First, let's compute $\(\frac{1}{x}\)$ :

$$
\(\frac{1}{x}=\frac{1}{-\frac{1}{98}}=-98\)
$$


Now, compute $\(x-\frac{1}{x}\)$ :

$$
\(x-\frac{1}{x}=-\frac{1}{98}-(-98)=-\frac{1}{98}+98=\frac{-1+98 \cdot 98}{98}=\frac{-1+9604}{98}=\frac{9603}{98}\)
$$


Next, compute $\(1-x\)$ :

$$\(1-x=1-\left(-\frac{1}{98}\right)=1+\frac{1}{98}=\frac{98}{98}+\frac{1}{98}=\frac{99}{98}\)$$


Now, the expression becomes:

$$
\(\frac{\frac{9603}{98}}{\frac{99}{98}}=\frac{9603}{98} \cdot \frac{98}{99}=\frac{9603}{99}\)
$$


Now, simplify $\(\frac{9603}{99}\)$ :
Let's perform the division:
$\(99 \times 97=9603\)$, because:
$\(99 \times 90=8910\)$
$\(99 \times 7=693\)$
$\(8910+693=9603\)$
So,

$$
\(\frac{9603}{99}=97\)
$$


Step 4: Verify the Answer
Looking back at the options:
A. -97
B. -90
C. 0
D. 90
E. 97

We found the value to be 97 , which corresponds to option E.