`lim_(xto-1) (1)/(sqrt(|x|-{-x}))` (where `{x}` denotes the fractional part of x) is equal to

Options

A. 16
B. 24
C. 32
D. 8

Explanation

To solve the limit \( \lim_{x \to -1} \frac{1}{\sqrt{|x| - \{x\}}} \), where \(\{x\}\) denotes the fractional part of \(x\), we will analyze the left-hand limit and the right-hand limit separately.

### Step 1: Understanding the fractional part function
The fractional part function \(\{x\}\) is defined as:
\[ \{x\} = x - \lfloor x \rfloor \]
For negative values of \(x\), \(\{x\} = x - (-n) = x + n\), where \(n\) is the integer part of \(x\).

### Step 2: Finding the left-hand limit
We will first find the left-hand limit as \(x\) approaches \(-1\) from the left (\(x \to -1^-\)):
\[ \lim_{x \to -1^-} \frac{1}{\sqrt{|x| - \{x\}}} \]

1. **Calculate \(|x|\)**: As \(x\) approaches \(-1\) from the left, \(|x| = -x\).
2. **Calculate \(\{x\}\)**: For \(x \to -1^-\), \(\{x\} = x + 1\).

Thus, we have:
\[ |x| - \{x\} = -x - (x + 1) = -x - x - 1 = -2x - 1 \]

### Step 3: Substitute into the limit
Now we substitute this back into the limit:
\[ \lim_{x \to -1^-} \frac{1}{\sqrt{-2x - 1}} \]

### Step 4: Evaluate the expression inside the limit
As \(x\) approaches \(-1\) from the left:
\[ -2x - 1 \to -2(-1) - 1 = 2 - 1 = 1 \]
Thus, we have:
\[ \sqrt{-2x - 1} \to \sqrt{1} = 1 \]

### Step 5: Calculate the left-hand limit
Now we can evaluate the left-hand limit:
\[ \lim_{x \to -1^-} \frac{1}{\sqrt{-2x - 1}} = \frac{1}{1} = 1 \]

### Step 6: Finding the right-hand limit
Next, we will find the right-hand limit as \(x\) approaches \(-1\) from the right (\(x \to -1^+\)):
\[ \lim_{x \to -1^+} \frac{1}{\sqrt{|x| - \{x\}}} \]

1. **Calculate \(|x|\)**: As \(x\) approaches \(-1\) from the right, \(|x| = -x\).
2. **Calculate \(\{x\}\)**: For \(x \to -1^+\), \(\{x\} = x + 1\).

Thus, we have:
\[ |x| - \{x\} = -x - (x + 1) = -x - x - 1 = -2x - 1 \]

### Step 7: Substitute into the limit
Now we substitute this back into the limit:
\[ \lim_{x \to -1^+} \frac{1}{\sqrt{-2x - 1}} \]

### Step 8: Evaluate the expression inside the limit
As \(x\) approaches \(-1\) from the right:
\[ -2x - 1 \to -2(-1) - 1 = 2 - 1 = 1 \]
Thus, we have:
\[ \sqrt{-2x - 1} \to \sqrt{1} = 1 \]

### Step 9: Calculate the right-hand limit
Now we can evaluate the right-hand limit:
\[ \lim_{x \to -1^+} \frac{1}{\sqrt{-2x - 1}} = \frac{1}{1} = 1 \]

### Conclusion
Since both the left-hand limit and the right-hand limit are equal:
\[ \lim_{x \to -1^-} \frac{1}{\sqrt{|x| - \{x\}}} = 1 \quad \text{and} \quad \lim_{x \to -1^+} \frac{1}{\sqrt{|x| - \{x\}}} = 1 \]
The overall limit exists and is equal to:
\[ \lim_{x \to -1} \frac{1}{\sqrt{|x| - \{x\}}} = 1 \]