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 can follow these steps:

### Step 1: Rewrite the limit
We start by rewriting the limit in a more manageable form:

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

### Step 2: Evaluate the first limit
Now, we will evaluate the first limit:

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

Using the standard limit result:

\[ \lim_{t \to 0} \frac{\log(1 + t)}{t} = 1, \]

we can substitute \( t = x^3 \). As \( x \to 0 \), \( t \to 0 \) as well. Thus,

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

### Step 3: Evaluate the second limit
Next, we evaluate the second limit:

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

Using the standard limit result:

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

we can rewrite this limit as:

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

### Step 4: Combine the results
Now we can combine the results from both limits:

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

### Final Answer
Thus, the limit is:

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

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