Evaluate the limits, if exist `lim_(xrarr0) (log(1+x^3))/(sin^3x)`

Explanation

To evaluate the limit

\[ \lim_{x \to 0} \frac{\log(1+x^3)}{\sin^3 x}, \]

we will follow these steps:

### Step 1: Identify the form of the limit
First, we substitute \( x = 0 \) into the expression:

\[ \log(1 + 0^3) = \log(1) = 0, \]
and
\[ \sin^3(0) = 0^3 = 0. \]

This gives us the indeterminate form \( \frac{0}{0} \).

**Hint:** When you encounter the \( \frac{0}{0} \) form, you can apply L'Hôpital's Rule or manipulate the expression to resolve the indeterminacy.

### Step 2: Rewrite the limit
We can rewrite the limit using known limits. We know that:

\[ \lim_{x \to 0} \frac{\log(1+x)}{x} = 1, \]
and
\[ \lim_{x \to 0} \frac{\sin x}{x} = 1. \]

We can manipulate our limit as follows:

\[ \lim_{x \to 0} \frac{\log(1+x^3)}{\sin^3 x} = \lim_{x \to 0} \frac{\log(1+x^3)}{x^3} \cdot \frac{x^3}{\sin^3 x}. \]

**Hint:** Break down the limit into parts that can be evaluated separately.

### Step 3: Evaluate the first part of the limit
Now we evaluate the first part:

\[ \lim_{x \to 0} \frac{\log(1+x^3)}{x^3}. \]

Using the known limit:

\[ \lim_{x \to 0} \frac{\log(1+x^3)}{x^3} = 1. \]

**Hint:** Substitute \( u = x^3 \) to convert the limit if necessary.

### Step 4: Evaluate the second part of the limit
Next, we evaluate the second part:

\[ \lim_{x \to 0} \frac{x^3}{\sin^3 x} = \left( \lim_{x \to 0} \frac{x}{\sin x} \right)^3. \]

Using the known limit:

\[ \lim_{x \to 0} \frac{x}{\sin x} = 1, \]

we find:

\[ \lim_{x \to 0} \frac{x^3}{\sin^3 x} = 1^3 = 1. \]

**Hint:** Remember that if \( \lim_{x \to 0} \frac{x}{\sin x} = 1 \), then raising it to any power will still yield 1.

### Step 5: Combine the results
Now we combine the results from both parts:

\[ \lim_{x \to 0} \frac{\log(1+x^3)}{\sin^3 x} = 1 \cdot 1 = 1. \]

### Final Answer
Thus, the limit is

\[ \boxed{1}. \]