`lim_(xrarr0^-)(sinx)/(sqrt(x))` is equal to

Options

A. 0
B. 1
C. `-(1)/(2)`
D. none of these

Answer

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

### Step 1: Rewrite the limit
We start with the limit expression:
\[ \lim_{x \to 0^-} \frac{\sin x}{\sqrt{x}} \]

### Step 2: Rationalize the denominator
To simplify the expression, we can multiply the numerator and denominator by \( \sqrt{x} \):
\[ \lim_{x \to 0^-} \frac{\sin x \cdot \sqrt{x}}{x} \]

### Step 3: Split the limit
We can separate the limit into two parts:
\[ \lim_{x \to 0^-} \frac{\sin x}{x} \cdot \lim_{x \to 0^-} \sqrt{x} \]

### Step 4: Evaluate the first limit
We know from standard limit results that:
\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \]
Thus:
\[ \lim_{x \to 0^-} \frac{\sin x}{x} = 1 \]

### Step 5: Evaluate the second limit
Next, we evaluate:
\[ \lim_{x \to 0^-} \sqrt{x} \]
As \( x \) approaches \( 0 \) from the left, \( \sqrt{x} \) approaches \( 0 \) (note that \( \sqrt{x} \) is not defined for negative \( x \), but as \( x \) approaches \( 0 \) from the negative side, we consider the limit of the function as it approaches \( 0 \)):
\[ \lim_{x \to 0^-} \sqrt{x} = 0 \]

### Step 6: Combine the results
Now, we combine the results of both limits:
\[ 1 \cdot 0 = 0 \]

### Final Answer
Thus, the limit is:
\[ \lim_{x \to 0^-} \frac{\sin x}{\sqrt{x}} = 0 \]

---