Question

If \( f \) is differentiable at \( x = 1 \), then \( \lim_{x \to 1} \frac{x^2 f(1) - f(x)}{x - 1} \) is

• A. \( -f'(1) \)
• B. \( f(1) - f'(1) \)
• C. \( 2f(1) - f'(1) \)
• D. \( 2f(1) + f'(1) \)
• E. \( f(1) + f'(1) \)

Hint:

Break expressions cleverly to introduce derivative definition.

Answer 1

The Correct Option is C

Concept: Use definition of derivative: \[ f'(1) = \lim_{x \to 1} \frac{f(x) - f(1)}{x-1} \]

Step 1: Rewrite expression.
\[ \frac{x^2 f(1) - f(x)}{x - 1} = \frac{x^2 f(1) - f(1) + f(1) - f(x)}{x-1} \]

Step 2: Split into two parts.
\[ = \frac{f(1)(x^2 -1)}{x-1} + \frac{f(1) - f(x)}{x-1} \]

Step 3: Simplify first term.
\[ x^2 -1 = (x-1)(x+1) \Rightarrow \frac{f(1)(x+1)(x-1)}{x-1} = f(1)(x+1) \]

Step 4: Take limit.
\[ \lim_{x\to1} f(1)(x+1) = 2f(1) \] Second term: \[ \lim_{x\to1} \frac{f(1)-f(x)}{x-1} = -f'(1) \]

Step 5: Combine.
\[ 2f(1) - f'(1) \]