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

Answer

To evaluate the limit

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

we need to analyze the behavior of the function as \( x \) approaches \( 0 \) from both the left and the right.

### Step 1: Evaluate the Right-Hand Limit (RHL)

When \( x \) approaches \( 0 \) from the right (i.e., \( x \to 0^+ \)), we have \( x > 0 \). Therefore, \( |x| = x \).

So,

\[ f(x) = \frac{|x|}{x} = \frac{x}{x} = 1 \quad \text{for } x > 0. \]

Thus,

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

### Step 2: Evaluate the Left-Hand Limit (LHL)

When \( x \) approaches \( 0 \) from the left (i.e., \( x \to 0^- \)), we have \( x < 0 \). Therefore, \( |x| = -x \).

So,

\[ f(x) = \frac{|x|}{x} = \frac{-x}{x} = -1 \quad \text{for } x < 0. \]

Thus,

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

### Step 3: Compare the Limits and the Function Value

Now we have:

- Right-Hand Limit: \( \lim_{x \to 0^+} f(x) = 1 \)
- Left-Hand Limit: \( \lim_{x \to 0^-} f(x) = -1 \)
- Value of the function at \( x = 0 \): \( f(0) = 0 \)

### Step 4: Conclusion

Since the right-hand limit and the left-hand limit are not equal (i.e., \( 1 \neq -1 \)), the overall limit does not exist.

Thus, we conclude that:

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