IDNLearn.com: Your trusted source for accurate and reliable answers. Join our knowledgeable community to find the answers you need for any topic or issue.
Sagot :
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.
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. IDNLearn.com has the answers you need. Thank you for visiting, and we look forward to helping you again soon.
