Calculate `lim_(x to 2) `, where `f(x) = {{:(3 if ,x le 2),(4 if, x gt 2):}`

Explanation

To solve the problem of finding the limit of the function \( f(x) \) as \( x \) approaches 2, we need to evaluate the left-hand limit (LHL) and the right-hand limit (RHL).

Given:
\[ f(x) = \begin{cases} 3 & \text{if } x \leq 2 \\ 4 & \text{if } x > 2 \end{cases} \]

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

The left-hand limit as \( x \) approaches 2 is given by:
\[ \lim_{x \to 2^-} f(x) \]
Since we are approaching from the left (values less than 2), we use the definition of \( f(x) \) for \( x \leq 2 \):
\[ f(x) = 3 \]
Thus,
\[ \lim_{x \to 2^-} f(x) = 3 \]

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

The right-hand limit as \( x \) approaches 2 is given by:
\[ \lim_{x \to 2^+} f(x) \]
Since we are approaching from the right (values greater than 2), we use the definition of \( f(x) \) for \( x > 2 \):
\[ f(x) = 4 \]
Thus,
\[ \lim_{x \to 2^+} f(x) = 4 \]

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

Now we compare the two limits:
- Left-Hand Limit (LHL) = 3
- Right-Hand Limit (RHL) = 4

Since the left-hand limit is not equal to the right-hand limit:
\[ \lim_{x \to 2^-} f(x) \neq \lim_{x \to 2^+} f(x) \]

### Conclusion

Since the left-hand limit and right-hand limit are not equal, the limit of \( f(x) \) as \( x \) approaches 2 does not exist:
\[ \lim_{x \to 2} f(x) \text{ does not exist.} \]

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