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

Options

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

Answer

To solve the limit problem given that \( f \) is derivable at \( x = a \), we need to evaluate the limit:

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

### Step 1: Substitute the limit
First, we substitute \( x = a \) into the expression:

\[ \frac{af(a) - af(a)}{a - a} = \frac{0}{0} \]

This results in an indeterminate form \( \frac{0}{0} \), which allows us to apply L'Hôpital's Rule.

### Step 2: Apply L'Hôpital's Rule
According to L'Hôpital's Rule, we differentiate the numerator and the denominator separately:

- **Numerator**: Differentiate \( xf(a) - af(x) \)
- The derivative of \( xf(a) \) with respect to \( x \) is \( f(a) \) (since \( f(a) \) is a constant).
- The derivative of \( -af(x) \) with respect to \( x \) is \( -af'(x) \).

Thus, the derivative of the numerator is:

\[ f(a) - af'(x) \]

- **Denominator**: Differentiate \( x - a \)

The derivative of \( x - a \) is simply \( 1 \).

### Step 3: Rewrite the limit
Now we can rewrite the limit using the derivatives we found:

\[ \lim_{x \to a} \frac{f(a) - af'(x)}{1} \]

### Step 4: Substitute \( x = a \) again
Now substitute \( x = a \) into the limit:

\[ f(a) - af'(a) \]

### Final Result
Thus, we conclude that:

\[ \lim_{x \to a} \frac{xf(a) - af(x)}{x - a} = f(a) - af'(a) \]