Find lim f(a) 2-2 Find lim f(a) 2-2 Find lim f(x) 2-2 Find f (2) Determine if the function is continuous or discontinuous at the lin indicate if the discontinuity is removable or non-removable. O The function is continuous at a = 2 O The function has a removable discontinuity at x = 2 O The function has a non-removable discontinuity at z = 2 If the function has a discontinuity at the limit value, check all the is discontinuous there. Of (2) does not exist Olim f(x) does not exist 2-2 Of (2) and lim f(x) both exist, but f (2) #lim f(x) 2-2 2-2
Find lim f(a) 2-2 Find lim f(a) 2-2 Find lim f(x) 2-2 Find f (2) Determine if the function is continuous or discontinuous at the lin indicate if the discontinuity is removable or non-removable. O The function is continuous at a = 2 O The function has a removable discontinuity at x = 2 O The function has a non-removable discontinuity at z = 2 If the function has a discontinuity at the limit value, check all the is discontinuous there. Of (2) does not exist Olim f(x) does not exist 2-2 Of (2) and lim f(x) both exist, but f (2) #lim f(x) 2-2 2-2
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter3: Functions
Section3.3: More On Functions; Piecewise-defined Functions
Problem 84E
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![### Calculus: Limits and Continuity
#### Question
Consider the function \( f(x) \) shown in the graph. Analyze the function around \( x = 2 \).
1. **Find the following limits:**
- \( \lim_{{x \to 2^-}} f(x) \) = [______]
- \( \lim_{{x \to 2^+}} f(x) \) = [______]
- \( \lim_{{x \to 2}} f(x) \) = [______]
2. **Find \( f(2) \):**
- \( f(2) \) = [______]
3. **Determine if the function is continuous or discontinuous at the limit value \( x = 2 \). Also, indicate if the discontinuity is removable or non-removable.**
- \( f(2) \) does not exist.
- \( \lim_{{x \to 2}} f(x) \) does not exist.
- \( f(2) \) and \( \lim_{{x \to 2}} f(x) \) both exist, but \( f(2) \neq \lim_{{x \to 2}} f(x) \).
#### Graph Description:
The graph of \( f(x) \) shows a parabola that opens upwards with a discontinuity at \( x = 2 \). There is a visible hole at \( (2, -2) \), indicating a removable discontinuity, and the function value at \( x = 2 \) is not defined on the graph.
#### Explanation:
- **Left-hand limit ( \( \lim_{{x \to 2^-}} f(x) \) ):** This is the value that the function approaches as \( x \) approaches 2 from the left.
- **Right-hand limit ( \( \lim_{{x \to 2^+}} f(x) \) ):** This is the value that the function approaches as \( x \) approaches 2 from the right.
- **Two-sided limit ( \( \lim_{{x \to 2}} f(x) \) ):** This is the value that the function approaches as \( x \) approaches 2 from both sides.
- **Value of the function at \( x = 2 \) ( \( f(2) \) ):**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcff5798b-7fe0-40b4-8b2c-b580bf1cfc79%2F709ea3db-4675-4ad0-bf66-9180ace2fc01%2Fqv99m7d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculus: Limits and Continuity
#### Question
Consider the function \( f(x) \) shown in the graph. Analyze the function around \( x = 2 \).
1. **Find the following limits:**
- \( \lim_{{x \to 2^-}} f(x) \) = [______]
- \( \lim_{{x \to 2^+}} f(x) \) = [______]
- \( \lim_{{x \to 2}} f(x) \) = [______]
2. **Find \( f(2) \):**
- \( f(2) \) = [______]
3. **Determine if the function is continuous or discontinuous at the limit value \( x = 2 \). Also, indicate if the discontinuity is removable or non-removable.**
- \( f(2) \) does not exist.
- \( \lim_{{x \to 2}} f(x) \) does not exist.
- \( f(2) \) and \( \lim_{{x \to 2}} f(x) \) both exist, but \( f(2) \neq \lim_{{x \to 2}} f(x) \).
#### Graph Description:
The graph of \( f(x) \) shows a parabola that opens upwards with a discontinuity at \( x = 2 \). There is a visible hole at \( (2, -2) \), indicating a removable discontinuity, and the function value at \( x = 2 \) is not defined on the graph.
#### Explanation:
- **Left-hand limit ( \( \lim_{{x \to 2^-}} f(x) \) ):** This is the value that the function approaches as \( x \) approaches 2 from the left.
- **Right-hand limit ( \( \lim_{{x \to 2^+}} f(x) \) ):** This is the value that the function approaches as \( x \) approaches 2 from the right.
- **Two-sided limit ( \( \lim_{{x \to 2}} f(x) \) ):** This is the value that the function approaches as \( x \) approaches 2 from both sides.
- **Value of the function at \( x = 2 \) ( \( f(2) \) ):**
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