Show that `Lim_( xto 2) (|x-2|)/(x-2)` does not exist .

Answer

To show that the limit \( \lim_{x \to 2} \frac{|x - 2|}{x - 2} \) does not exist, we will calculate the left-hand limit and the right-hand limit as \( x \) approaches 2.

### Step 1: Define the Left-Hand Limit
We start by calculating the left-hand limit, which is denoted as:
\[ \lim_{x \to 2^-} \frac{|x - 2|}{x - 2} \]
For values of \( x \) that are less than 2, \( |x - 2| \) can be expressed as:
\[ |x - 2| = -(x - 2) = 2 - x \]
Thus, we can rewrite the limit as:
\[ \lim_{x \to 2^-} \frac{2 - x}{x - 2} \]
This simplifies to:
\[ \lim_{x \to 2^-} \frac{-(x - 2)}{x - 2} = \lim_{x \to 2^-} -1 = -1 \]

### Step 2: Define the Right-Hand Limit
Next, we calculate the right-hand limit, which is denoted as:
\[ \lim_{x \to 2^+} \frac{|x - 2|}{x - 2} \]
For values of \( x \) that are greater than 2, \( |x - 2| \) can be expressed as:
\[ |x - 2| = x - 2 \]
Thus, we can rewrite the limit as:
\[ \lim_{x \to 2^+} \frac{x - 2}{x - 2} \]
This simplifies to:
\[ \lim_{x \to 2^+} 1 = 1 \]

### Step 3: Compare the Left-Hand and Right-Hand Limits
Now we compare the two limits we calculated:
- Left-hand limit: \( -1 \)
- Right-hand limit: \( 1 \)

Since the left-hand limit and the right-hand limit are not equal:
\[ -1 \neq 1 \]
we conclude that the overall limit does not exist.

### Conclusion
Thus, we can state that:
\[ \lim_{x \to 2} \frac{|x - 2|}{x - 2} \text{ does not exist.} \]

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