Question

If \( f(x) \) is differentiable at \( x = 1 \) and \[ \lim_{h \to 0} \frac{1}{h} f(1 + h) = 5, \] then \( f'(1) \) is equal to:

• A. \( 6 \)
• B. \( 5 \)
• C. \( 4 \)
• D. \( 3 \)

Hint:

The derivative \( f'(x) \) represents the instantaneous rate of change of the function at \( x \). When working with limits, carefully analyze the continuity and differentiability conditions.

Answer 1

The Correct Option is B

The derivative \( f'(1) \) is defined as:\[f'(1) = \lim_{h \to 0} \frac{f(1 + h) - f(1)}{h}.\]Given the condition:\[\lim_{h \to 0} \frac{1}{h} f(1 + h) = 5.\]Decomposing the limit expression:\[\lim_{h \to 0} \frac{1}{h} f(1 + h) = \lim_{h \to 0} \left( \frac{f(1 + h) - f(1)}{h} + \frac{f(1)}{h} \right).\]For the limit to be finite, \( \frac{f(1)}{h} \) must approach 0 as \( h \to 0 \). This requires:\[f(1) = 0.\]Substituting \( f(1) = 0 \) into the equation:\[\lim_{h \to 0} \frac{1}{h} f(1 + h) = \lim_{h \to 0} \frac{f(1 + h) - f(1)}{h} = f'(1).\]Therefore:\[f'(1) = 5.\] Final Answer:\[\boxed{5}\]