If `f(x)={{:(,2x+3, x le 0),(,3(x+1), x gt 0):}` then the value of `underset(x to 0)lim f(x)` is

Options

A. 0
B. 6
C. 2
D. 3

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:
\[ f(x) = \begin{cases} 2x + 3 & \text{if } x \leq 0 \\ 3(x + 1) & \text{if } x > 0 \end{cases} \]

### Step 1: Calculate the left-hand limit as \( x \) approaches 0
The left-hand limit is denoted as:
\[ \lim_{x \to 0^-} f(x) \]
Since we are approaching 0 from the left (where \( x \leq 0 \)), we use the first piece of the function:
\[ f(x) = 2x + 3 \]
Now, substituting \( x = 0 \):
\[ \lim_{x \to 0^-} f(x) = 2(0) + 3 = 3 \]

### Step 2: Calculate the right-hand limit as \( x \) approaches 0
The right-hand limit is denoted as:
\[ \lim_{x \to 0^+} f(x) \]
Since we are approaching 0 from the right (where \( x > 0 \)), we use the second piece of the function:
\[ f(x) = 3(x + 1) \]
Now, substituting \( x = 0 \):
\[ \lim_{x \to 0^+} f(x) = 3(0 + 1) = 3 \]

### Step 3: Compare the left-hand limit and right-hand limit
Now we compare both limits:
\[ \lim_{x \to 0^-} f(x) = 3 \quad \text{and} \quad \lim_{x \to 0^+} f(x) = 3 \]
Since both limits are equal, we can conclude:
\[ \lim_{x \to 0} f(x) = 3 \]

### Final Answer
Thus, the value of \( \lim_{x \to 0} f(x) \) is \( 3 \).

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