Evaluate `lim_(hto0)(log_(10)(1+h))/h`

Answer

To evaluate the limit \( \lim_{h \to 0} \frac{\log_{10}(1+h)}{h} \), we can follow these steps:

### Step 1: Identify the form of the limit
When we substitute \( h = 0 \) into the expression, we get:
\[ \frac{\log_{10}(1+0)}{0} = \frac{\log_{10}(1)}{0} = \frac{0}{0} \]
This is an indeterminate form, so we can apply L'Hôpital's Rule.

### Step 2: Apply L'Hôpital's Rule
L'Hôpital's Rule states that if we have an indeterminate form of type \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), we can differentiate the numerator and the denominator separately.

The numerator is \( \log_{10}(1+h) \) and the denominator is \( h \).

#### Differentiate the numerator:
Using the chain rule, the derivative of \( \log_{10}(1+h) \) is:
\[ \frac{d}{dh} \log_{10}(1+h) = \frac{1}{(1+h) \ln(10)} \]
where \( \ln(10) \) is the natural logarithm of 10.

#### Differentiate the denominator:
The derivative of \( h \) is simply:
\[ \frac{d}{dh} h = 1 \]

### Step 3: Rewrite the limit using the derivatives
Now, we can rewrite the limit using the derivatives:
\[ \lim_{h \to 0} \frac{\log_{10}(1+h)}{h} = \lim_{h \to 0} \frac{\frac{1}{(1+h) \ln(10)}}{1} \]

### Step 4: Evaluate the limit
Now we can substitute \( h = 0 \) into the expression:
\[ \lim_{h \to 0} \frac{1}{(1+h) \ln(10)} = \frac{1}{(1+0) \ln(10)} = \frac{1}{\ln(10)} \]

### Final Result
Thus, the limit is:
\[ \lim_{h \to 0} \frac{\log_{10}(1+h)}{h} = \frac{1}{\ln(10)} \]