Prove that `Lim_(x to 0) (|x|)/x , x != 0 ` does not exist .

Answer

To prove that \(\lim_{x \to 0} \frac{|x|}{x}\) does not exist, we will evaluate the left-hand limit and the right-hand limit separately.

### Step 1: Evaluate the Right-Hand Limit
We start by finding the right-hand limit as \(x\) approaches 0 from the positive side.

\[ \lim_{x \to 0^+} \frac{|x|}{x} \]

Since \(x\) is positive when approaching from the right, we have \(|x| = x\). Thus, we can rewrite the limit as:

\[ \lim_{x \to 0^+} \frac{x}{x} = \lim_{x \to 0^+} 1 = 1 \]

### Step 2: Evaluate the Left-Hand Limit
Next, we find the left-hand limit as \(x\) approaches 0 from the negative side.

\[ \lim_{x \to 0^-} \frac{|x|}{x} \]

Since \(x\) is negative when approaching from the left, we have \(|x| = -x\). Thus, we can rewrite the limit as:

\[ \lim_{x \to 0^-} \frac{-x}{x} = \lim_{x \to 0^-} -1 = -1 \]

### Step 3: Compare the Left-Hand and Right-Hand Limits
Now we compare the left-hand limit and the right-hand limit:

- Right-hand limit: \(1\)
- Left-hand limit: \(-1\)

Since the left-hand limit and the right-hand limit are not equal:

\[ \lim_{x \to 0^+} \frac{|x|}{x} \neq \lim_{x \to 0^-} \frac{|x|}{x} \]

### Conclusion
Therefore, we conclude that:

\[ \lim_{x \to 0} \frac{|x|}{x} \text{ does not exist.} \]

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