If f is derivable at x =a,then ` underset(xto a ) lim( (xf(a) -af( x))/(x-a) ) `

Options

A. f'(a)
B. af'(a) - f(a)
C. f(a) - af'(a)
D. af'(a)

Answer

To solve the problem, we need to evaluate the limit:

\[ \lim_{x \to a} \frac{x f(a) - a f(x)}{x - a} \]

Given that \( f \) is derivable at \( x = a \), we can utilize the definition of the derivative and some algebraic manipulation.

### Step-by-Step Solution:

1. **Start with the limit expression**:
\[ L = \lim_{x \to a} \frac{x f(a) - a f(x)}{x - a} \]

2. **Rewrite the expression**:
We can factor out \( f(a) \) from the first term:
\[ L = \lim_{x \to a} \frac{f(a)(x - a) + a(f(a) - f(x))}{x - a} \]

3. **Separate the limit**:
This can be split into two parts:
\[ L = \lim_{x \to a} \left( f(a) + \frac{a(f(a) - f(x))}{x - a} \right) \]

4. **Evaluate the first part**:
As \( x \to a \), the first part simplifies to:
\[ f(a) \]

5. **Evaluate the second part**:
The second part involves the derivative of \( f \) at \( a \):
\[ \lim_{x \to a} \frac{f(a) - f(x)}{x - a} = -f'(a) \]
Thus, we have:
\[ L = f(a) + a(-f'(a)) = f(a) - a f'(a) \]

6. **Final result**:
Therefore, the limit evaluates to:
\[ L = f(a) - a f'(a) \]

### Conclusion:
Thus, the final answer is:
\[ \lim_{x \to a} \frac{x f(a) - a f(x)}{x - a} = f(a) - a f'(a) \]