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: $$f'(1) = \lim_{h \to 0} \frac{f(1 + h) - f(1)}{h}.$$ We are given the following condition: $$\lim_{h \to 0} \frac{1}{h} f(1 + h) = 5.$$ Next, decompose 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 exist and be finite, it must be true that $\frac{f(1)}{h} \to 0$ as $h \to 0$. This implies: $$f(1) = 0.$$ Now substitute $f(1) = 0$ into the equation: $$\lim_{h \to 0} \frac{1}{h} f(1 + h) = \lim_{h \to 0} \frac{f(1 + h) - f(1)}{h} = f'(1).$$ Thus, we find: $$f'(1) = 5.$$ Final Answer: $$\boxed{5}$$