If `f(1) =g(1)=2`, then `lim_(xrarr1) (f(1)g(x)-f(x)g(1)-f(1)+g(1))/(f(x)-g(x))` is equal to

Options

A. 0
B. 1
C. 2
D. -2

Answer

To solve the limit problem, we start with the expression given:

\[ \lim_{x \to 1} \frac{f(1)g(x) - f(x)g(1) - f(1) + g(1)}{f(x) - g(x)} \]

Given that \(f(1) = g(1) = 2\), we can substitute these values into the expression.

### Step 1: Substitute the values
Substituting \(f(1)\) and \(g(1)\) into the limit gives:

\[ \lim_{x \to 1} \frac{2g(x) - f(x) \cdot 2 - 2 + 2}{f(x) - g(x)} \]

This simplifies to:

\[ \lim_{x \to 1} \frac{2g(x) - 2f(x)}{f(x) - g(x)} \]

### Step 2: Factor out common terms
We can factor out 2 from the numerator:

\[ \lim_{x \to 1} \frac{2(g(x) - f(x))}{f(x) - g(x)} \]

### Step 3: Simplify the expression
Notice that \(f(x) - g(x) = -(g(x) - f(x))\). Thus, we can rewrite the limit as:

\[ \lim_{x \to 1} \frac{2(g(x) - f(x))}{-(g(x) - f(x))} \]

This simplifies to:

\[ \lim_{x \to 1} -2 = -2 \]

### Conclusion
Thus, the final answer is:

\[ \boxed{-2} \]