Suppose f(x) is differentibale for all x and `lim_(h to 0) (1)/(h) (1+h)=5"then f'(1) equals"`

Options

A. 6
B. 5
C. 4
D. 3

Explanation

To solve the problem step by step, we start with the given information and apply the definition of the derivative.

### Step-by-Step Solution:

1. **Understanding the Limit**: We are given that
\[ \lim_{h \to 0} \frac{1}{h}(1 + h) = 5. \]
We can simplify this limit:
\[ \lim_{h \to 0} \frac{1 + h}{h} = \lim_{h \to 0} \left( \frac{1}{h} + 1 \right). \]
As \( h \to 0 \), \( \frac{1}{h} \) approaches infinity, which implies that \( f(1) \) must be finite for the limit to equal 5.

2. **Using the Definition of the Derivative**: The derivative of \( f \) at \( x = 1 \) is defined as:
\[ f'(1) = \lim_{h \to 0} \frac{f(1 + h) - f(1)}{h}. \]

3. **Rearranging the Limit**: We can express the limit in terms of the given limit:
\[ f'(1) = \lim_{h \to 0} \left( \frac{1}{h}(1 + h) - \frac{f(1)}{h} \right). \]
This can be rewritten as:
\[ f'(1) = \lim_{h \to 0} \left( \frac{1 + h - f(1)}{h} \right). \]

4. **Substituting the Limit**: From the original limit, we know that:
\[ \lim_{h \to 0} \frac{1 + h}{h} = 5. \]
Therefore, we can substitute this into our expression:
\[ f'(1) = 5 - \lim_{h \to 0} \frac{f(1)}{h}. \]

5. **Analyzing the Finite Quantity**: Since \( f'(1) \) must be finite, the term \( \lim_{h \to 0} \frac{f(1)}{h} \) must also be finite. This is only possible if \( f(1) = 0 \).

6. **Finding \( f'(1) \)**: If \( f(1) = 0 \), we substitute back into our equation:
\[ f'(1) = 5 - 0 = 5. \]

### Conclusion:
Thus, we find that
\[ f'(1) = 5. \]