Evaluate the following limits :
`Lim_(x to 0) (x^(3))/(sin x^(2))`

Answer

To evaluate the limit \( \lim_{x \to 0} \frac{x^3}{\sin(x^2)} \), we can follow these steps:

### Step 1: Rewrite the limit
We can rewrite the limit as:
\[ \lim_{x \to 0} \frac{x^3}{\sin(x^2)} = \lim_{x \to 0} \frac{x^2}{\frac{\sin(x^2)}{x}} \]

### Step 2: Factor out \(x^2\)
Now, we can separate the limit into two parts:
\[ \lim_{x \to 0} \frac{x^3}{\sin(x^2)} = \lim_{x \to 0} x^2 \cdot \lim_{x \to 0} \frac{1}{\frac{\sin(x^2)}{x^2}} \]

### Step 3: Use the limit property
We know from the standard limit that:
\[ \lim_{u \to 0} \frac{\sin(u)}{u} = 1 \]
In our case, let \(u = x^2\). As \(x \to 0\), \(u \to 0\) as well. Thus:
\[ \lim_{x \to 0} \frac{\sin(x^2)}{x^2} = 1 \]
This implies:
\[ \lim_{x \to 0} \frac{x^2}{\sin(x^2)} = 1 \]

### Step 4: Combine the limits
Substituting back into our limit, we have:
\[ \lim_{x \to 0} \frac{x^3}{\sin(x^2)} = \lim_{x \to 0} x^2 \cdot \lim_{x \to 0} \frac{1}{\frac{\sin(x^2)}{x^2}} = \lim_{x \to 0} x^2 \cdot 1 = \lim_{x \to 0} x^2 \]

### Step 5: Evaluate the limit
Now, we can evaluate:
\[ \lim_{x \to 0} x^2 = 0 \]

### Final Answer
Thus, the limit is:
\[ \lim_{x \to 0} \frac{x^3}{\sin(x^2)} = 0 \]