Answer :
To determine the limit of the given piecewise function [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\(-2\)[/tex], we need to examine the behavior of the function from both the left ([tex]\( x \to -2^- \)[/tex]) and the right ([tex]\( x \to -2^+ \)[/tex]). Additionally, we need to consider the value of the function at [tex]\( x = -2 \)[/tex].
The given function is:
[tex]\[ f(x) = \begin{cases} 3 - x^2 & \text{if } x < -2 \\ 0 & \text{if } x = -2 \\ 11 - x^2 & \text{if } x > -2 \end{cases} \][/tex]
### Step 1: Calculate the Left-Hand Limit ([tex]\( x \to -2^- \)[/tex])
When [tex]\( x \)[/tex] approaches [tex]\(-2\)[/tex] from the left, we use the expression for [tex]\( x < -2 \)[/tex]:
[tex]\[ f(x) = 3 - x^2 \][/tex]
We need to find:
[tex]\[ \lim_{x \to -2^-} f(x) = \lim_{x \to -2^-} (3 - x^2) \][/tex]
Substituting [tex]\(-2\)[/tex] into this expression:
[tex]\[ 3 - (-2)^2 = 3 - 4 = -1 \][/tex]
Thus, the left-hand limit is:
[tex]\[ \lim_{x \to -2^-} f(x) = -1 \][/tex]
### Step 2: Calculate the Right-Hand Limit ([tex]\( x \to -2^+ \)[/tex])
When [tex]\( x \)[/tex] approaches [tex]\(-2\)[/tex] from the right, we use the expression for [tex]\( x > -2 \)[/tex]:
[tex]\[ f(x) = 11 - x^2 \][/tex]
We need to find:
[tex]\[ \lim_{x \to -2^+} f(x) = \lim_{x \to -2^+} (11 - x^2) \][/tex]
Substituting [tex]\(-2\)[/tex] into this expression:
[tex]\[ 11 - (-2)^2 = 11 - 4 = 7 \][/tex]
Thus, the right-hand limit is:
[tex]\[ \lim_{x \to -2^+} f(x) = 7 \][/tex]
### Step 3: Examine the value at [tex]\( x = -2 \)[/tex]
The value of [tex]\( f(x) \)[/tex] at [tex]\( x = -2 \)[/tex] is given directly:
[tex]\[ f(-2) = 0 \][/tex]
### Step 4: Interpret the Results
To determine if the limit exists and equals 7, the left-hand limit and the right-hand limit must both equal 7. From the calculations, we have:
[tex]\[ \lim_{x \to -2^-} f(x) = -1 \quad \text{and} \quad \lim_{x \to -2^+} f(x) = 7 \][/tex]
Additionally, the value at [tex]\( x = -2 \)[/tex] is 0, but this does not necessarily affect the limit itself.
Since the left-hand limit ([tex]\(-1\)[/tex]) and the right-hand limit ([tex]\(7\)[/tex]) are not equal, the general limit [tex]\( \lim_{x \to -2} f(x) \)[/tex] does not exist.
In summary:
- Left-hand limit: [tex]\( -1 \)[/tex]
- Right-hand limit: [tex]\( 7 \)[/tex]
- Value at [tex]\( x = -2 \)[/tex]: Undefined (or [tex]\(0\)[/tex] if interpreted directly from the function definition)
Thus, [tex]\( \lim_{x \to -2} f(x) \)[/tex] does not equal 7 because the left and right-hand limits are different. Additionally, check for necessity conditions:
- The left-hand limit does not satisfy the condition [tex]\( \lim_{x \to -2^-} f(x) = 7 \)[/tex].
- The right-hand limit satisfies [tex]\( \lim_{x \to -2^+} f(x) = 7 \)[/tex].
Hence, the correct interpretation is:
- Left-hand condition not satisfied.
- Right-hand condition satisfied.
The given function is:
[tex]\[ f(x) = \begin{cases} 3 - x^2 & \text{if } x < -2 \\ 0 & \text{if } x = -2 \\ 11 - x^2 & \text{if } x > -2 \end{cases} \][/tex]
### Step 1: Calculate the Left-Hand Limit ([tex]\( x \to -2^- \)[/tex])
When [tex]\( x \)[/tex] approaches [tex]\(-2\)[/tex] from the left, we use the expression for [tex]\( x < -2 \)[/tex]:
[tex]\[ f(x) = 3 - x^2 \][/tex]
We need to find:
[tex]\[ \lim_{x \to -2^-} f(x) = \lim_{x \to -2^-} (3 - x^2) \][/tex]
Substituting [tex]\(-2\)[/tex] into this expression:
[tex]\[ 3 - (-2)^2 = 3 - 4 = -1 \][/tex]
Thus, the left-hand limit is:
[tex]\[ \lim_{x \to -2^-} f(x) = -1 \][/tex]
### Step 2: Calculate the Right-Hand Limit ([tex]\( x \to -2^+ \)[/tex])
When [tex]\( x \)[/tex] approaches [tex]\(-2\)[/tex] from the right, we use the expression for [tex]\( x > -2 \)[/tex]:
[tex]\[ f(x) = 11 - x^2 \][/tex]
We need to find:
[tex]\[ \lim_{x \to -2^+} f(x) = \lim_{x \to -2^+} (11 - x^2) \][/tex]
Substituting [tex]\(-2\)[/tex] into this expression:
[tex]\[ 11 - (-2)^2 = 11 - 4 = 7 \][/tex]
Thus, the right-hand limit is:
[tex]\[ \lim_{x \to -2^+} f(x) = 7 \][/tex]
### Step 3: Examine the value at [tex]\( x = -2 \)[/tex]
The value of [tex]\( f(x) \)[/tex] at [tex]\( x = -2 \)[/tex] is given directly:
[tex]\[ f(-2) = 0 \][/tex]
### Step 4: Interpret the Results
To determine if the limit exists and equals 7, the left-hand limit and the right-hand limit must both equal 7. From the calculations, we have:
[tex]\[ \lim_{x \to -2^-} f(x) = -1 \quad \text{and} \quad \lim_{x \to -2^+} f(x) = 7 \][/tex]
Additionally, the value at [tex]\( x = -2 \)[/tex] is 0, but this does not necessarily affect the limit itself.
Since the left-hand limit ([tex]\(-1\)[/tex]) and the right-hand limit ([tex]\(7\)[/tex]) are not equal, the general limit [tex]\( \lim_{x \to -2} f(x) \)[/tex] does not exist.
In summary:
- Left-hand limit: [tex]\( -1 \)[/tex]
- Right-hand limit: [tex]\( 7 \)[/tex]
- Value at [tex]\( x = -2 \)[/tex]: Undefined (or [tex]\(0\)[/tex] if interpreted directly from the function definition)
Thus, [tex]\( \lim_{x \to -2} f(x) \)[/tex] does not equal 7 because the left and right-hand limits are different. Additionally, check for necessity conditions:
- The left-hand limit does not satisfy the condition [tex]\( \lim_{x \to -2^-} f(x) = 7 \)[/tex].
- The right-hand limit satisfies [tex]\( \lim_{x \to -2^+} f(x) = 7 \)[/tex].
Hence, the correct interpretation is:
- Left-hand condition not satisfied.
- Right-hand condition satisfied.
