Discuss the continuity of the function
`f(x)={((|x|)/x", " xne 0),(1", " x=0):}` at x=0

Answer

To discuss the continuity of the function

\[ f(x) = \begin{cases} \frac{|x|}{x} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases} \]

at \(x = 0\), we need to check the left-hand limit (LHL), right-hand limit (RHL), and the value of the function at that point.

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

The left-hand limit as \(x\) approaches 0 from the left (denoted as \(0^-\)) is given by:

\[ \text{LHL} = \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} \frac{|x|}{x} \]

Since \(x\) is approaching 0 from the left, \(x\) is negative. Therefore, \(|x| = -x\). Thus, we have:

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

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

The right-hand limit as \(x\) approaches 0 from the right (denoted as \(0^+\)) is given by:

\[ \text{RHL} = \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} \frac{|x|}{x} \]

Since \(x\) is approaching 0 from the right, \(x\) is positive. Therefore, \(|x| = x\). Thus, we have:

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

### Step 3: Value of the Function at \(x = 0\)

The value of the function at \(x = 0\) is given directly by the function definition:

\[ f(0) = 1 \]

### Step 4: Compare LHL, RHL, and \(f(0)\)

Now we compare the limits and the function value:

- LHL = -1
- RHL = 1
- \(f(0) = 1\)

Since LHL \(\neq\) RHL, we conclude that the function is not continuous at \(x = 0\).

### Conclusion

Thus, the function \(f(x)\) is discontinuous at \(x = 0\).

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