`lim_(x to 0) (1 - cos x)/(x sqrt(x^(2))`

Options

A. `(1)/(2)`
B. `-(1)/(2)`
C. 0
D. Does not exist

Answer

To solve the limit problem \( \lim_{x \to 0} \frac{1 - \cos x}{x \sqrt{x^2}} \), we can follow these steps:

### Step 1: Simplify the Expression
We start with the limit:
\[ \lim_{x \to 0} \frac{1 - \cos x}{x \sqrt{x^2}} \]
The term \( \sqrt{x^2} \) can be simplified. Since \( \sqrt{x^2} = |x| \), we rewrite the expression:
\[ \lim_{x \to 0} \frac{1 - \cos x}{x |x|} \]

### Step 2: Consider the Absolute Value
Now, we need to consider the absolute value \( |x| \):
- For \( x \to 0^+ \) (approaching from the right), \( |x| = x \).
- For \( x \to 0^- \) (approaching from the left), \( |x| = -x \).

### Step 3: Calculate the Left-Hand Limit
We first calculate the left-hand limit as \( x \to 0^- \):
\[ \lim_{x \to 0^-} \frac{1 - \cos x}{x (-x)} = \lim_{x \to 0^-} \frac{1 - \cos x}{-x^2} \]
Using the known limit \( \lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2} \), we have:
\[ \lim_{x \to 0^-} \frac{1 - \cos x}{-x^2} = -\frac{1}{2} \]

### Step 4: Calculate the Right-Hand Limit
Now we calculate the right-hand limit as \( x \to 0^+ \):
\[ \lim_{x \to 0^+} \frac{1 - \cos x}{x x} = \lim_{x \to 0^+} \frac{1 - \cos x}{x^2} \]
Using the same known limit:
\[ \lim_{x \to 0^+} \frac{1 - \cos x}{x^2} = \frac{1}{2} \]

### Step 5: Compare the Limits
Now we compare the two limits:
- Left-hand limit: \( -\frac{1}{2} \)
- Right-hand limit: \( \frac{1}{2} \)

Since the left-hand limit does not equal the right-hand limit, we conclude that:
\[ \lim_{x \to 0} \frac{1 - \cos x}{x \sqrt{x^2}} \text{ does not exist.} \]

### Final Answer
Thus, the limit does not exist.