If `f(x) = (|X|)/(X)`. Discuss the continuity at `x rarr 0`

Answer

To determine the continuity of the function \( f(x) = \frac{|x|}{x} \) at \( x = 0 \), we need to check the left-hand limit and the right-hand limit as \( x \) approaches 0.

### Step 1: Calculate the Left-Hand Limit
The left-hand limit is defined as:
\[ \lim_{x \to 0^-} f(x) \]
When \( x \) approaches 0 from the left (i.e., \( x < 0 \)), the absolute value function behaves as follows:
\[ |x| = -x \]
Thus, we can rewrite \( f(x) \) for \( x < 0 \):
\[ f(x) = \frac{|x|}{x} = \frac{-x}{x} = -1 \]
Now we can find the left-hand limit:
\[ \lim_{x \to 0^-} f(x) = -1 \]

### Step 2: Calculate the Right-Hand Limit
The right-hand limit is defined as:
\[ \lim_{x \to 0^+} f(x) \]
When \( x \) approaches 0 from the right (i.e., \( x > 0 \)), the absolute value function behaves as follows:
\[ |x| = x \]
Thus, we can rewrite \( f(x) \) for \( x > 0 \):
\[ f(x) = \frac{|x|}{x} = \frac{x}{x} = 1 \]
Now we can find the right-hand limit:
\[ \lim_{x \to 0^+} f(x) = 1 \]

### Step 3: Compare the Limits
Now we compare the left-hand limit and the right-hand limit:
\[ \lim_{x \to 0^-} f(x) = -1 \quad \text{and} \quad \lim_{x \to 0^+} f(x) = 1 \]
Since the left-hand limit does not equal the right-hand limit:
\[ -1 \neq 1 \]

### Conclusion
Since the left-hand limit and the right-hand limit at \( x = 0 \) are not equal, the function \( f(x) = \frac{|x|}{x} \) is not continuous at \( x = 0 \).

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