Evaluate the following limits :
`Lim_(x to 0) (log(1-x/2))/x `

Answer

To evaluate the limit

\[ \lim_{x \to 0} \frac{\log(1 - \frac{x}{2})}{x}, \]

we will follow these steps:

### Step 1: Substitute \( x = 0 \)

First, we substitute \( x = 0 \) into the limit:

\[ \frac{\log(1 - \frac{0}{2})}{0} = \frac{\log(1)}{0} = \frac{0}{0}. \]

This is an indeterminate form (0/0), so we can apply L'Hôpital's Rule.

### Step 2: Apply L'Hôpital's Rule

According to L'Hôpital's Rule, when we have an indeterminate form of type \( \frac{0}{0} \), we can take the derivative of the numerator and the derivative of the denominator separately.

The numerator is \( \log(1 - \frac{x}{2}) \) and the denominator is \( x \).

- Derivative of the numerator:

Using the chain rule, we have:

\[ \frac{d}{dx} \log(1 - \frac{x}{2}) = \frac{1}{1 - \frac{x}{2}} \cdot \left(-\frac{1}{2}\right) = -\frac{1}{2(1 - \frac{x}{2})}. \]

- Derivative of the denominator:

The derivative of \( x \) is simply \( 1 \).

So, applying L'Hôpital's Rule gives us:

\[ \lim_{x \to 0} \frac{\log(1 - \frac{x}{2})}{x} = \lim_{x \to 0} \frac{-\frac{1}{2(1 - \frac{x}{2})}}{1} = \lim_{x \to 0} -\frac{1}{2(1 - \frac{x}{2})}. \]

### Step 3: Evaluate the limit

Now we substitute \( x = 0 \) into the expression:

\[ -\frac{1}{2(1 - \frac{0}{2})} = -\frac{1}{2(1)} = -\frac{1}{2}. \]

### Final Result

Thus, the limit is:

\[ \lim_{x \to 0} \frac{\log(1 - \frac{x}{2})}{x} = -\frac{1}{2}. \]

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