Evaluate: `lim_(xto1)(sin|||x|-2|-3|)` is

Options

A. `sin 2`
B. `sin 1`
C. ` 0`
D. Does not exist

Answer

To evaluate the limit \( \lim_{x \to 1} \sin \left| \left| |x| - 2 \right| - 3 \right| \), we can break down the expression step by step.

### Step 1: Substitute \( x = 1 \)
First, we substitute \( x = 1 \) into the expression to see if we can directly evaluate the limit.

\[ |x| = |1| = 1 \]

### Step 2: Evaluate the inner absolute value
Now we evaluate \( ||x| - 2| \):

\[ ||1| - 2| = |1 - 2| = |-1| = 1 \]

### Step 3: Evaluate the next absolute value
Next, we evaluate \( |||x| - 2| - 3| \):

\[ ||1 - 3| = |1 - 3| = |-2| = 2 \]

### Step 4: Evaluate the sine function
Now we substitute this value into the sine function:

\[ \sin \left| \left| |x| - 2 \right| - 3 \right| = \sin(2) \]

### Step 5: Conclude the limit
Since we have evaluated the expression without encountering any indeterminate forms, we can conclude:

\[ \lim_{x \to 1} \sin \left| \left| |x| - 2 \right| - 3 \right| = \sin(2) \]

Thus, the final answer is:

\[ \sin(2) \]