Limits and Continuity Consider the graphed piecewise function f (x) shown below. • Determine the open intervals on which f (x) is continuous. • Determine, for the values x = -3,, x = 1, and x = 2, for which values f (x) is: o Continuous; o Left-continuous; o Right-continuous. For each value that f (x) has a discontinuity, state: o Which type of discontinuity it is; o Which of the three criteria for continuity are and are not met; o Why the limit criterion specifically is or is not met. • Determine whether f(x) has one or more horizontal asymptotes, using limits at ∞ and -∞ in your explanation. ● -8- -6- -4-
Limits and Continuity Consider the graphed piecewise function f (x) shown below. • Determine the open intervals on which f (x) is continuous. • Determine, for the values x = -3,, x = 1, and x = 2, for which values f (x) is: o Continuous; o Left-continuous; o Right-continuous. For each value that f (x) has a discontinuity, state: o Which type of discontinuity it is; o Which of the three criteria for continuity are and are not met; o Why the limit criterion specifically is or is not met. • Determine whether f(x) has one or more horizontal asymptotes, using limits at ∞ and -∞ in your explanation. ● -8- -6- -4-
Chapter2: Functions And Their Graphs
Section2.4: A Library Of Parent Functions
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Transcribed Image Text:# Limits and Continuity
Consider the graphed piecewise function \( f(x) \) shown below.
- Determine the open intervals on which \( f(x) \) is continuous.
- Determine, for the values \( x = -3 \), \( x = 1 \), and \( x = 2 \), for which values \( f(x) \) is:
- Continuous;
- Left-continuous;
- Right-continuous.
- For each value that \( f(x) \) has a discontinuity, state:
- Which type of discontinuity it is;
- Which of the three criteria for continuity are and are not met;
- Why the limit criterion specifically is or is not met.
- Determine whether \( f(x) \) has one or more horizontal asymptotes, using limits at \(\infty\) and \(-\infty\) in your explanation.
## Graph Explanation
The graph displays a piecewise function \( f(x) \) with several distinct segments and behaviors:
- For \( x < -2 \), a purple curve approximately follows a horizontal pattern around \( y = 4 \). The endpoint at \( x = -3 \) is represented by an open circle, indicating the function is not defined at \( x = -3 \).
- For \( -2 < x < 0 \), there's a decreasing red segment. The segment ends at \( x = 0 \) with an open circle at \( y = 2 \).
- For \( 0 < x < 2 \), two segments are visible:
- A green segment approaches a vertical asymptote as \( x \) approaches 0 from the right, indicating the function heads towards infinity.
- A blue curve starts at \( (0, 6) \) and increases to a value just below \( y = 8 \) at \( x = 1 \). There is an open circle at \( (1, 8) \), indicating the function is not defined at this point.
- For \( x > 2 \), a green curve shows the function f(x) approaches 0 as \( x \) increases without bound, depicting a horizontal asymptote at \( y = 0 \).
Each segment demonstrates various continuous and discontinuous behaviors, which include:
- Removable discontinuities (open circles indicating undefined points),
- Jump discontinuities (different function values
![### Function Definition and Analysis through Limits and Continuity
Consider the equation for the function \( f(x) \) given by:
\[
f(x) =
\begin{cases}
e^{x+2} + 4, & x < -3 \\
\sqrt{x+3} + 1, & -3 \le x < 1 \\
2, & x = 1 \\
3x, & 1 < x < 2 \\
-\ln(x-2) + 3, & x > 2
\end{cases}
\]
Use the equation along with the limit definitions of one-sided and two-sided continuity to confirm the results you obtained from the graph regarding continuity and discontinuity at each point. After that, apply the limits at positive and negative infinity to verify the end behavior of the function.
### Steps to Analyze the Function:
**1. Check Continuity at Key Points:**
- Evaluate the function at the boundaries where the definition changes: \( x = -3, x = 1, \) and \( x = 2 \).
- Verify if \( \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a) \) at each boundary point for continuity.
**2. Evaluate Limits at Infinity:**
- Compute \( \lim_ {x \to \infty} f(x) \) and \( \lim_{x \to -\infty} f(x) \).
By following these steps, students can gain a comprehensive understanding of how to apply the limit definitions to investigate a piecewise function's continuity and end behavior.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F36ade1a1-f50c-49e6-9973-e1b61dc80212%2F38d66cd9-d928-4275-8ed2-ec0177d003a5%2Fxswxe1_processed.png&w=3840&q=75)
Transcribed Image Text:### Function Definition and Analysis through Limits and Continuity
Consider the equation for the function \( f(x) \) given by:
\[
f(x) =
\begin{cases}
e^{x+2} + 4, & x < -3 \\
\sqrt{x+3} + 1, & -3 \le x < 1 \\
2, & x = 1 \\
3x, & 1 < x < 2 \\
-\ln(x-2) + 3, & x > 2
\end{cases}
\]
Use the equation along with the limit definitions of one-sided and two-sided continuity to confirm the results you obtained from the graph regarding continuity and discontinuity at each point. After that, apply the limits at positive and negative infinity to verify the end behavior of the function.
### Steps to Analyze the Function:
**1. Check Continuity at Key Points:**
- Evaluate the function at the boundaries where the definition changes: \( x = -3, x = 1, \) and \( x = 2 \).
- Verify if \( \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a) \) at each boundary point for continuity.
**2. Evaluate Limits at Infinity:**
- Compute \( \lim_ {x \to \infty} f(x) \) and \( \lim_{x \to -\infty} f(x) \).
By following these steps, students can gain a comprehensive understanding of how to apply the limit definitions to investigate a piecewise function's continuity and end behavior.
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