If `f(x) = {{:(x, x lt 0),(1, x=0),(x^(2), x gt a):}`, then `lim_(x to 0) f(x)=`

Options

A. 0
B. 1
C. 2
D. not exist

Answer

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 separately.

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

### Step 1: Calculate the Left-Hand Limit
The left-hand limit as \( x \) approaches 0 is given by:
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} x \]
Since \( x < 0 \) in this case, we substitute \( x \) with values approaching 0 from the left:
\[ \lim_{x \to 0^-} x = 0 \]

### Step 2: Calculate the Right-Hand Limit
The right-hand limit as \( x \) approaches 0 is given by:
\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x^2 \]
Since \( x > 0 \) in this case, we substitute \( x \) with values approaching 0 from the right:
\[ \lim_{x \to 0^+} x^2 = 0^2 = 0 \]

### Step 3: Evaluate the Function at \( x = 0 \)
The value of the function at \( x = 0 \) is:
\[ f(0) = 1 \]

### Step 4: Check for Limit Existence
For the limit to exist, the left-hand limit, right-hand limit, and the function value at that point must all be equal:
- Left-hand limit: \( 0 \)
- Right-hand limit: \( 0 \)
- Function value at \( 0 \): \( 1 \)

Since the left-hand limit (0) is not equal to the function value (1), we conclude that the limit does not exist.

### Final Answer
\[ \lim_{x \to 0} f(x) \text{ does not exist.} \]

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