If `f(x)={{:(x","" "xlt0),(1","" "x=0),(x^(2)","" "xgt0):}," then find " lim_(xto0) f(x)"` if exists.

Explanation

To find the limit of the function \( f(x) \) as \( x \) approaches 0, we need to evaluate the left-hand limit and the right-hand limit at that point. The function is defined as follows:

\[ f(x) = \begin{cases} x & \text{if } x < 0 \\ 1 & \text{if } x = 0 \\ x^2 & \text{if } x > 0 \end{cases} \]

### Step 1: Evaluate the Left-Hand Limit
The left-hand limit as \( x \) approaches 0 is given by:

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

For \( x < 0 \), the function \( f(x) = x \). Therefore, we can compute:

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

### Step 2: Evaluate the Right-Hand Limit
The right-hand limit as \( x \) approaches 0 is given by:

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

For \( x > 0 \), the function \( f(x) = x^2 \). Therefore, we can compute:

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

### Step 3: Compare the Left-Hand and Right-Hand Limits
Now we compare the two limits we calculated:

- Left-hand limit: \( \lim_{x \to 0^-} f(x) = 0 \)
- Right-hand limit: \( \lim_{x \to 0^+} f(x) = 0 \)

Since both limits are equal, we can conclude that:

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

### Final Result
Thus, the limit is:

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

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