Does `Lim_(x to 0 ) f(x)` exist if f(x) = `{:{(x," when "x lt 0 ),(0 ," when " x = 0 ),(x^(2)," when " x gt 0):}`

Answer

To determine whether the limit \(\lim_{x \to 0} f(x)\) exists for the given piecewise function:

\[ f(x) = \begin{cases} x & \text{when } x < 0 \\ 0 & \text{when } x = 0 \\ x^2 & \text{when } x > 0 \end{cases} \]

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

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

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

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

Since \(f(x) = x\) when \(x < 0\), we can substitute:

\[ \lim_{x \to 0^-} x = 0 \]

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

The right-hand limit as \(x\) approaches 0 is given by:

\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x^2 \]

Since \(f(x) = x^2\) when \(x > 0\), we can substitute:

\[ \lim_{x \to 0^+} x^2 = 0 \]

### Step 3: Evaluate \(f(0)\)

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

\[ f(0) = 0 \]

### Step 4: Compare the Limits and Function Value

Now we have:

- Left-Hand Limit (LHL): \(0\)
- Right-Hand Limit (RHL): \(0\)
- Value at \(x = 0\): \(f(0) = 0\)

Since both the left-hand limit and the right-hand limit are equal and also equal to \(f(0)\):

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

### Conclusion

Thus, we conclude that:

\[ \lim_{x \to 0} f(x) \text{ exists and is equal to } 0. \]

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