Let f(x) = [x] and [] represents the greatest integer function, then

Options

A. `underset( x to 1 +)(lim f(x) = 1`
B. `underset(x to 2008-)(lim) f(x) = 2007`
C. `underset(x to k)(lim) f(x) = 2008 , K in (2008, 2009)`
D. `underset(x to 2008-)(lim) x - [x] = 1`

Answer

To solve the problem, we need to evaluate the limits involving the greatest integer function \( f(x) = [x] \), where \( [x] \) denotes the greatest integer less than or equal to \( x \). We will analyze the limits step by step.

### Step 1: Evaluate \( \lim_{x \to 1^+} f(x) \)

1. **Definition**: The greatest integer function \( f(x) = [x] \) gives the largest integer less than or equal to \( x \).
2. **Limit from the right**: As \( x \) approaches 1 from the right (i.e., \( x = 1 + h \) where \( h \to 0^+ \)), we have:
\[ f(1 + h) = [1 + h] = 1 \quad \text{(since \( 1 + h \) is still less than 2)} \]
3. **Conclusion**: Therefore,
\[ \lim_{x \to 1^+} f(x) = 1 \]

### Step 2: Evaluate \( \lim_{x \to 2008^-} f(x) \)

1. **Limit from the left**: As \( x \) approaches 2008 from the left (i.e., \( x = 2008 - h \) where \( h \to 0^+ \)), we have:
\[ f(2008 - h) = [2008 - h] = 2007 \quad \text{(since \( 2008 - h \) is less than 2008)} \]
2. **Conclusion**: Therefore,
\[ \lim_{x \to 2008^-} f(x) = 2007 \]

### Step 3: Evaluate \( \lim_{x \to K} f(x) \) where \( K \in (2008, 2009) \)

1. **Choosing \( K \)**: Since \( K \) is between 2008 and 2009, we can take \( K = 2008 + h \) where \( h \to 0^+ \).
2. **Value of the function**: For \( K \) in this interval,
\[ f(K) = [K] = [2008 + h] = 2008 \quad \text{(since \( K \) is still less than 2009)} \]
3. **Conclusion**: Therefore,
\[ \lim_{x \to K} f(x) = 2008 \]

### Step 4: Evaluate \( \lim_{x \to 2008^-} (x - f(x)) \)

1. **Expression**: We need to evaluate \( \lim_{x \to 2008^-} (x - f(x)) \).
2. **Substituting \( x \)**: As \( x \) approaches 2008 from the left,
\[ f(x) = 2007 \quad \text{(as established in Step 2)} \]
Thus,
\[ x - f(x) = x - 2007 \]
3. **Limit calculation**: Therefore,
\[ \lim_{x \to 2008^-} (x - f(x)) = \lim_{x \to 2008^-} (x - 2007) = 2008 - 2007 = 1 \]

### Summary of Results
1. \( \lim_{x \to 1^+} f(x) = 1 \)
2. \( \lim_{x \to 2008^-} f(x) = 2007 \)
3. \( \lim_{x \to K} f(x) = 2008 \) for \( K \in (2008, 2009) \)
4. \( \lim_{x \to 2008^-} (x - f(x)) = 1 \)