Evaluate the following :
`lim_(x to 0)(sinx^(@))/x`

Answer

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

### Step 1: Rewrite the limit
We start with the limit:
\[ \lim_{x \to 0} \frac{\sin(x^\theta)}{x} \]

### Step 2: Use the substitution for small angles
As \( x \to 0 \), \( x^\theta \) also approaches 0. We can use the fact that \( \lim_{y \to 0} \frac{\sin(y)}{y} = 1 \). To apply this, we rewrite the limit in terms of \( x^\theta \):
\[ \lim_{x \to 0} \frac{\sin(x^\theta)}{x^\theta} \cdot \frac{x^\theta}{x} \]

### Step 3: Simplify the expression
Now, we can simplify \( \frac{x^\theta}{x} \):
\[ \frac{x^\theta}{x} = x^{\theta - 1} \]
Thus, our limit becomes:
\[ \lim_{x \to 0} \frac{\sin(x^\theta)}{x^\theta} \cdot x^{\theta - 1} \]

### Step 4: Apply the limit
Now, we can evaluate the limit:
\[ \lim_{x \to 0} \frac{\sin(x^\theta)}{x^\theta} = 1 \]
So we have:
\[ \lim_{x \to 0} \left( 1 \cdot x^{\theta - 1} \right) = \lim_{x \to 0} x^{\theta - 1} \]

### Step 5: Determine the final limit based on \( \theta \)
- If \( \theta > 1 \), then \( \lim_{x \to 0} x^{\theta - 1} = 0 \).
- If \( \theta = 1 \), then \( \lim_{x \to 0} x^{\theta - 1} = \lim_{x \to 0} x^0 = 1 \).
- If \( \theta < 1 \), then \( \lim_{x \to 0} x^{\theta - 1} = \infty \).

### Final Answer
Thus, the limit evaluates to:
\[ \lim_{x \to 0} \frac{\sin(x^\theta)}{x} = \begin{cases} 0 & \text{if } \theta > 1 \\ 1 & \text{if } \theta = 1 \\ \infty & \text{if } \theta < 1 \end{cases} \]