`Lt_(x to 0) (sin sqrt(theta))/sqrt(sin theta)`=

Options

A. 1
B. `-1`
C. 0
D. none

Answer

To solve the limit \( \lim_{\theta \to 0} \frac{\sin(\sqrt{\theta})}{\sqrt{\sin(\theta)}} \), we will follow these steps:

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

### Step 2: Separate the Limit
We can separate the limit into two parts:
\[ L = \lim_{\theta \to 0} \frac{\sin(\sqrt{\theta})}{\sqrt{\theta}} \cdot \lim_{\theta \to 0} \frac{\sqrt{\theta}}{\sqrt{\sin(\theta)}} \]

### Step 3: Evaluate the First Limit
The first limit can be evaluated using the standard limit:
\[ \lim_{x \to 0} \frac{\sin(x)}{x} = 1 \]
Here, we substitute \( x = \sqrt{\theta} \), so as \( \theta \to 0 \), \( \sqrt{\theta} \to 0 \):
\[ \lim_{\theta \to 0} \frac{\sin(\sqrt{\theta})}{\sqrt{\theta}} = 1 \]

### Step 4: Evaluate the Second Limit
For the second limit, we use the fact that as \( \theta \to 0 \), \( \sin(\theta) \) behaves like \( \theta \):
\[ \lim_{\theta \to 0} \frac{\sqrt{\theta}}{\sqrt{\sin(\theta)}} = \lim_{\theta \to 0} \sqrt{\frac{\theta}{\sin(\theta)}} \]
Using the standard limit \( \lim_{\theta \to 0} \frac{\theta}{\sin(\theta)} = 1 \):
\[ \lim_{\theta \to 0} \sqrt{\frac{\theta}{\sin(\theta)}} = \sqrt{1} = 1 \]

### Step 5: Combine the Results
Now we combine both limits:
\[ L = 1 \cdot 1 = 1 \]

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

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