Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
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-

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question

Please help! 

# 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
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.
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.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 4 steps with 18 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS