If the function `f (x) = {{:(3,x lt 0),(12, x gt 0):}` then `lim_(x to 0) f (x) =`

Options

A. 0
B. 3
C. 12
D. Does not exist

Answer

To solve the problem, we need to find the limit of the function \( f(x) \) as \( x \) approaches 0. The function is defined piecewise as follows:

\[ f(x) = \begin{cases} 3 & \text{if } x < 0 \\ 12 & \text{if } x > 0 \end{cases} \]

### Step 1: Find the Right-Hand Limit (RHL)

The right-hand limit as \( x \) approaches 0 (denoted as \( \lim_{x \to 0^+} f(x) \)) considers values of \( x \) that are greater than 0.

Since \( f(x) = 12 \) when \( x > 0 \):

\[ \lim_{x \to 0^+} f(x) = 12 \]

### Step 2: Find the Left-Hand Limit (LHL)

The left-hand limit as \( x \) approaches 0 (denoted as \( \lim_{x \to 0^-} f(x) \)) considers values of \( x \) that are less than 0.

Since \( f(x) = 3 \) when \( x < 0 \):

\[ \lim_{x \to 0^-} f(x) = 3 \]

### Step 3: Compare the Left-Hand Limit and Right-Hand Limit

Now we compare the two limits we found:

- Right-Hand Limit (RHL): \( \lim_{x \to 0^+} f(x) = 12 \)
- Left-Hand Limit (LHL): \( \lim_{x \to 0^-} f(x) = 3 \)

Since the left-hand limit (3) is not equal to the right-hand limit (12), we conclude that the overall limit does not exist.

### Final Answer

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

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