Question

Let $f(x)$ be a differentiable function at $x = a$ with $f'(a) = 4$ and $f(a) = 8$. Then, $\lim_{x\to a} \frac{xf(a) - af(x)}{x - a} = $_______

• A. $4(1 - 2a)$
• B. $4(2a - 1)$
• C. $4(2 - a)$
• D. $4(a - 2)$

Hint:

An alternative algebraic approach is to add and subtract $a f(a)$ in the numerator:
\[ \lim_{x \to a} \frac{x f(a) - a f(a) + a f(a) - a f(x)}{x - a} = \lim_{x \to a} \left[ f(a) \frac{x - a}{x - a} - a \frac{f(x) - f(a)}{x - a} \right] \]
\[ = f(a) - a \lim_{x \to a} \frac{f(x) - f(a)}{x - a} = f(a) - a f'(a) \]
This matches the definition of the derivative and avoids using L'Hopital's rule.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This question asks us to compute the limit of a functional expression as $x$ approaches $a$, using the given values of the function $f(a)$ and its derivative $f'(a)$.

Step 2: Key Formula or Approach:
Substituting $x = a$ directly into the limit expression yields the indeterminate form $\frac{0}{0}$.
Therefore, we can apply L'Hopital's rule, which states that the limit of a ratio of two functions is equal to the limit of the ratio of their derivatives:
\[ \lim_{x \to a} \frac{g(x)}{h(x)} = \lim_{x \to a} \frac{g'(x)}{h'(x)} \]

Step 3: Detailed Explanation:

• The given values are:
\[ f(a) = 8, \quad f'(a) = 4 \]

• The limit to evaluate is:
\[ L = \lim_{x\to a} \frac{x f(a) - a f(x)}{x - a} \]

• Direct substitution of $x = a$ gives:
\[ \frac{a f(a) - a f(a)}{a - a} = \frac{0}{0} \]

• Since this is an indeterminate form, we apply L'Hopital's rule by differentiating the numerator and the denominator with respect to $x$:

• The derivative of the numerator with respect to $x$ is:
\[ \frac{d}{dx} [x f(a) - a f(x)] = f(a) \cdot 1 - a \cdot f'(x) = f(a) - a f'(x) \]

• The derivative of the denominator with respect to $x$ is:
\[ \frac{d}{dx} [x - a] = 1 \]

• Now, we compute the limit of the ratio of these derivatives:
\[ L = \lim_{x \to a} \frac{f(a) - a f'(x)}{1} \]

• Since $f'(x)$ is continuous at $x = a$ (as $f(x)$ is differentiable there), we can substitute $x = a$ directly into the derivative expression:
\[ L = f(a) - a f'(a) \]

• Substitute the given values $f(a) = 8$ and $f'(a) = 4$:
\[ L = 8 - a(4) = 8 - 4a \]

• Factor out 4 from the expression:
\[ L = 4(2 - a) \]

Step 4: Final Answer
The value of the limit is $4(2 - a)$, which corresponds to option (C).