Evaluate `lim_(x to 0)f(x)`, where :
`" "f(x)={{:(x/abs(x)",", x ne 0), (0",", x=0):}`

Answer

To evaluate the limit \( \lim_{x \to 0} f(x) \), where

\[ f(x) = \begin{cases} \frac{x}{|x|} & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \]

we will analyze the left-hand limit (LHL) and the right-hand limit (RHL) as \( x \) approaches 0.

### Step 1: Calculate the Left-Hand Limit (LHL)

The left-hand limit is defined as:

\[ \lim_{x \to 0^-} f(x) \]

For \( x < 0 \), we have \( |x| = -x \). Thus,

\[ f(x) = \frac{x}{|x|} = \frac{x}{-x} = -1 \]

Therefore,

\[ \lim_{x \to 0^-} f(x) = -1 \]

### Step 2: Calculate the Right-Hand Limit (RHL)

The right-hand limit is defined as:

\[ \lim_{x \to 0^+} f(x) \]

For \( x > 0 \), we have \( |x| = x \). Thus,

\[ f(x) = \frac{x}{|x|} = \frac{x}{x} = 1 \]

Therefore,

\[ \lim_{x \to 0^+} f(x) = 1 \]

### Step 3: Compare LHL and RHL

Now we compare the two limits:

- LHL as \( x \to 0^- \) is \(-1\)
- RHL as \( x \to 0^+ \) is \(1\)

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

\[ \lim_{x \to 0^-} f(x) \neq \lim_{x \to 0^+} f(x) \]

### Conclusion

The limit \( \lim_{x \to 0} f(x) \) does not exist because the left-hand limit and the right-hand limit are not equal.

Thus, the final answer is:

\[ \lim_{x \to 0} f(x) \text{ does not exist.} \]

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