To determine the values of [tex]\( a \)[/tex] for which [tex]\( \lim_{x \to a} f(x) \)[/tex] exists, we need to consider the points where the function [tex]\( f(x) \)[/tex] changes its definition. These points are [tex]\(-1\)[/tex] and [tex]\(2\)[/tex]. At these points, we need to ensure that the limit from the left is equal to the limit from the right.
Let's evaluate the limits at these points step-by-step:
### 1. Evaluating the limit as [tex]\( x \)[/tex] approaches [tex]\(-1\)[/tex]
For [tex]\( x < -1 \)[/tex], the function is given by [tex]\( f(x) = 2 - x \)[/tex].
For [tex]\( -1 \leq x < 2 \)[/tex], the function is given by [tex]\( f(x) = x \)[/tex].
- Limit from the left of [tex]\(-1\)[/tex] ([tex]\(x \to -1^-\)[/tex]):
[tex]\[
\lim_{x \to -1^-} f(x) = \lim_{x \to -1^-} (2 - x)
\][/tex]
Substituting [tex]\( x = -1 \)[/tex]:
[tex]\[
\lim_{x \to -1^-} (2 - x) = 2 - (-1) = 3
\][/tex]
- Limit from the right of [tex]\(-1\)[/tex] ([tex]\(x \to -1^+\)[/tex]):
[tex]\[
\lim_{x \to -1^+} f(x) = \lim_{x \to -1^+} x
\][/tex]
Substituting [tex]\( x = -1 \)[/tex]:
[tex]\[
\lim_{x \to -1^+} x = -1
\][/tex]
Since the limits from the left and right at [tex]\( x = -1 \)[/tex] are not equal ([tex]\( 3 \neq -1 \)[/tex]), [tex]\( \lim_{x \to -1} f(x) \)[/tex] does not exist.
### 2. Evaluating the limit as [tex]\( x \)[/tex] approaches [tex]\( 2 \)[/tex]
For [tex]\( -1 \leq x < 2 \)[/tex], the function is given by [tex]\( f(x) = x \)[/tex].
For [tex]\( x \geq 2 \)[/tex], the function is given by [tex]\( f(x) = (x - 2)^2 \)[/tex].
- Limit from the left of [tex]\( 2 \)[/tex] ([tex]\(x \to 2^-\)[/tex]):
[tex]\[
\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} x
\][/tex]
Substituting [tex]\( x = 2 \)[/tex]:
[tex]\[
\lim_{x \to 2^-} x = 2
\][/tex]
- Limit from the right of [tex]\( 2 \)[/tex] ([tex]\(x \to 2^+\)[/tex]):
[tex]\[
\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (x - 2)^2
\][/tex]
Substituting [tex]\( x = 2 \)[/tex]:
[tex]\[
\lim_{x \to 2^+} (x - 2)^2 = (2 - 2)^2 = 0
\][/tex]
Since the limits from the left and right at [tex]\( x = 2 \)[/tex] are not equal ([tex]\( 2 \neq 0 \)[/tex]), [tex]\( \lim_{x \to 2} f(x) \)[/tex] does not exist.
### Conclusion
After analyzing the limits at the boundary points [tex]\(-1\)[/tex] and [tex]\(2\)[/tex]:
- The limit as [tex]\( x \)[/tex] approaches [tex]\(-1\)[/tex] does not exist.
- The limit as [tex]\( x \)[/tex] approaches [tex]\( 2 \)[/tex] does not exist.
Therefore, there are no values of [tex]\( a \)[/tex] for which [tex]\( \lim_{x \to a} f(x) \)[/tex] exists.
