Evaluate : `lim_(x to 0)(log(1+x))/sinx`.

Answer

To evaluate the limit \( \lim_{x \to 0} \frac{\log(1+x)}{\sin x} \), we can follow these steps:

### Step 1: Rewrite the limit
We start with the limit:
\[ L = \lim_{x \to 0} \frac{\log(1+x)}{\sin x} \]

### Step 2: Use known limits
We know from calculus that:
\[ \lim_{x \to 0} \frac{\log(1+x)}{x} = 1 \]
and
\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \]

### Step 3: Rewrite the limit using these known limits
We can manipulate our limit \( L \) to use these known limits. We can multiply and divide by \( x \):
\[ L = \lim_{x \to 0} \frac{\log(1+x)}{\sin x} \cdot \frac{x}{x} = \lim_{x \to 0} \frac{\log(1+x)}{x} \cdot \frac{x}{\sin x} \]

### Step 4: Break it into two separate limits
Now we can separate the limit into two parts:
\[ L = \left( \lim_{x \to 0} \frac{\log(1+x)}{x} \right) \cdot \left( \lim_{x \to 0} \frac{x}{\sin x} \right) \]

### Step 5: Evaluate each limit
From our known limits:
1. \( \lim_{x \to 0} \frac{\log(1+x)}{x} = 1 \)
2. \( \lim_{x \to 0} \frac{x}{\sin x} = 1 \)

### Step 6: Combine the results
Now we can combine these results:
\[ L = 1 \cdot 1 = 1 \]

### Final Answer
Thus, the limit is:
\[ \lim_{x \to 0} \frac{\log(1+x)}{\sin x} = 1 \]

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