If `f(a)=2` ` f'(a)=1` ` g(a)=-1` `g'(a)=2` then `lim_(xtoa)(g(x)f(a)-g(a)f(x))/(x-a)` is

Options

A. `-5`
B. `3`
C. `-3`
D. `5`

Answer

To solve the limit problem given, we will use the properties of derivatives and apply L'Hôpital's Rule. Let's break down the solution step by step.

### Step-by-step Solution:

1. **Identify the limit expression:**
We need to evaluate:
\[ \lim_{x \to a} \frac{g(x)f(a) - g(a)f(x)}{x - a} \]

2. **Recognize the form of the limit:**
As \( x \to a \), both the numerator and denominator approach 0, which gives us the indeterminate form \( \frac{0}{0} \). Therefore, we can apply L'Hôpital's Rule.

3. **Differentiate the numerator and denominator:**
According to L'Hôpital's Rule, we differentiate the numerator and the denominator:
\[ \text{Numerator: } \frac{d}{dx}[g(x)f(a) - g(a)f(x)] = g'(x)f(a) - g(a)f'(x) \]
\[ \text{Denominator: } \frac{d}{dx}[x - a] = 1 \]

4. **Rewrite the limit using derivatives:**
Now we can rewrite the limit as:
\[ \lim_{x \to a} \left( g'(x)f(a) - g(a)f'(x) \right) \]

5. **Substitute \( x = a \):**
Now we substitute \( x = a \) into the expression:
\[ g'(a)f(a) - g(a)f'(a) \]

6. **Plug in the given values:**
We have the following values from the problem:
- \( f(a) = 2 \)
- \( f'(a) = 1 \)
- \( g(a) = -1 \)
- \( g'(a) = 2 \)

Substituting these values gives:
\[ = g'(a)f(a) - g(a)f'(a) = 2 \cdot 2 - (-1) \cdot 1 \]
\[ = 4 + 1 = 5 \]

7. **Final result:**
Thus, the limit evaluates to:
\[ \lim_{x \to a} \frac{g(x)f(a) - g(a)f(x)}{x - a} = 5 \]

### Summary of the Solution:
The limit is equal to 5.