Discuss the continuity of the function
`f(x) ={:{((Sinx)/(x) ", "x lt0 ),(x+1 ", " x ge0):},`
at =0.

Answer

To determine the continuity of the function

\[ f(x) = \begin{cases} \frac{\sin x}{x} & \text{if } x < 0 \\ x + 1 & \text{if } x \geq 0 \end{cases} \]

at \(x = 0\), we need to check three conditions:

1. The left-hand limit as \(x\) approaches \(0\).
2. The right-hand limit as \(x\) approaches \(0\).
3. The value of the function at \(x = 0\).

### Step 1: Calculate the value of the function at \(x = 0\)

Since \(f(x) = x + 1\) for \(x \geq 0\):

\[ f(0) = 0 + 1 = 1 \]

### Step 2: Calculate the left-hand limit as \(x\) approaches \(0\)

We need to find:

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

Using the known limit:

\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \]

Thus,

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

### Step 3: Calculate the right-hand limit as \(x\) approaches \(0\)

We need to find:

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

Substituting \(x = 0\):

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

### Step 4: Check the continuity condition

For \(f(x)\) to be continuous at \(x = 0\), the following must hold:

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

From our calculations:

- Left-hand limit: \(1\)
- Right-hand limit: \(1\)
- Value of the function at \(0\): \(1\)

Since all three values are equal, we conclude that:

\[ f(x) \text{ is continuous at } x = 0. \]

### Summary

The function \(f(x)\) is continuous at \(x = 0\) because:

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