If f(x) = `{:{(x/(|x|)" for "x != 0 ),(0 " for " x = 0 ):}` , find `Lim_(x to 0 ) f(x)`

Answer

To find the limit of the function \( f(x) \) as \( x \) approaches 0, we will evaluate the left-hand limit and the right-hand limit separately.

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

### Step 1: Find the Left-Hand Limit as \( x \) approaches 0

We denote the left-hand limit as:
\[ \lim_{x \to 0^-} f(x) \]

For \( x \) approaching 0 from the left, we can substitute \( x = -h \) where \( h \to 0^+ \). Thus, we have:
\[ \lim_{x \to 0^-} f(x) = \lim_{h \to 0^+} f(-h) \]

Now substituting into the function:
\[ f(-h) = \frac{-h}{|-h|} = \frac{-h}{h} = -1 \]

Thus, the left-hand limit is:
\[ \lim_{x \to 0^-} f(x) = -1 \]

### Step 2: Find the Right-Hand Limit as \( x \) approaches 0

Now we denote the right-hand limit as:
\[ \lim_{x \to 0^+} f(x) \]

For \( x \) approaching 0 from the right, we can substitute \( x = h \) where \( h \to 0^+ \). Thus, we have:
\[ \lim_{x \to 0^+} f(x) = \lim_{h \to 0^+} f(h) \]

Now substituting into the function:
\[ f(h) = \frac{h}{|h|} = \frac{h}{h} = 1 \]

Thus, the right-hand limit is:
\[ \lim_{x \to 0^+} f(x) = 1 \]

### Step 3: Compare the Left-Hand Limit and Right-Hand Limit

Now we compare the two limits:
- Left-hand limit: \( -1 \)
- Right-hand limit: \( 1 \)

Since the left-hand limit and right-hand limit are not equal:
\[ \lim_{x \to 0^-} f(x) \neq \lim_{x \to 0^+} f(x) \]

### Conclusion

Since the left-hand limit and right-hand limit are not equal, we conclude that:
\[ \lim_{x \to 0} f(x) \text{ does not exist.} \]