The value of `lim_(xrarr2^-) {x+(x-[x])^2}`, is

Options

A. 0
B. 1
C. 2
D. 3

Answer

To find the limit \( \lim_{x \to 2^-} \left( x + (x - [x])^2 \right) \), we need to evaluate the left-hand limit as \( x \) approaches 2 from the left.

### Step-by-Step Solution:

1. **Substitute \( x \) with \( 2 - h \)**:
Since we are approaching 2 from the left, we can express \( x \) as \( 2 - h \), where \( h \) is a small positive number approaching 0. Thus, we have:
\[ \lim_{h \to 0^+} \left( (2 - h) + ((2 - h) - [2 - h])^2 \right) \]

2. **Evaluate \( [2 - h] \)**:
The greatest integer function \( [x] \) gives the largest integer less than or equal to \( x \). For \( 2 - h \) (where \( h \) is a small positive number), \( [2 - h] = 1 \) because \( 2 - h \) is slightly less than 2.

3. **Calculate \( (2 - h) - [2 - h] \)**:
Now we can substitute \( [2 - h] \):
\[ (2 - h) - [2 - h] = (2 - h) - 1 = 1 - h \]

4. **Square the result**:
Now we square the result:
\[ (1 - h)^2 = 1 - 2h + h^2 \]

5. **Combine the terms**:
Substitute back into the limit expression:
\[ \lim_{h \to 0^+} \left( (2 - h) + (1 - 2h + h^2) \right) \]
Simplifying this gives:
\[ \lim_{h \to 0^+} \left( 2 - h + 1 - 2h + h^2 \right) = \lim_{h \to 0^+} \left( 3 - 3h + h^2 \right) \]

6. **Evaluate the limit**:
As \( h \) approaches 0, the expression simplifies to:
\[ 3 - 3(0) + (0)^2 = 3 \]

Thus, the value of the limit is:
\[ \boxed{3} \]