Homework Help is Here – Start Your Trial Now!
Math
Calculus
Let f(x) = 3x7 if x < 5 if 755 x ≥ 5 Show that f(x) has a jump discontinuity at x = 5 by calculating the limits from the left and right at x = 5. lim f(x) = 8 x-5- lim f(x) = -30 X

Let f(x) = 3x7 if x < 5 if 755 x ≥ 5 Show that f(x) has a jump discontinuity at x = 5 by calculating the limits from the left and right at x = 5. lim f(x) = 8 x-5- lim f(x) = -30 X

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
See similar textbooks
icon
Related questions
Question
**Piecewise Function Analysis and Discontinuity** Consider the piecewise function \( f(x) \): \[ f(x) = \begin{cases} 3x - 7 & \text{if } x < 5 \\ \frac{5}{x+5} & \text{if } x \geq 5 \end{cases} \] To demonstrate that \( f(x) \) has a jump discontinuity at \( x = 5 \), we calculate the limits from the left and right at this point. 1. **Left-hand Limit:** \(\lim_{x \to 5^-} f(x)\) Evaluating using \(3x - 7\): \[ = 3(5) - 7 = 15 - 7 = 8 \] *Result:* \( \lim_{x \to 5^-} f(x) = 8 \) (Correct) 2. **Right-hand Limit:** \(\lim_{x \to 5^+} f(x)\) Evaluating using \(\frac{5}{x+5}\): \[ = \frac{5}{5+5} = \frac{5}{10} = 0.5 \] Note: The image incorrectly shows \(-30\) as the value, which is a mistake. The discrepancy between the left-hand limit and right-hand limit at \( x = 5 \) illustrates a jump discontinuity, confirming that \( f(x) \) is not continuous at this point.
Transcribed Image Text:**Piecewise Function Analysis and Discontinuity** Consider the piecewise function \( f(x) \): \[ f(x) = \begin{cases} 3x - 7 & \text{if } x < 5 \\ \frac{5}{x+5} & \text{if } x \geq 5 \end{cases} \] To demonstrate that \( f(x) \) has a jump discontinuity at \( x = 5 \), we calculate the limits from the left and right at this point. 1. **Left-hand Limit:** \(\lim_{x \to 5^-} f(x)\) Evaluating using \(3x - 7\): \[ = 3(5) - 7 = 15 - 7 = 8 \] *Result:* \( \lim_{x \to 5^-} f(x) = 8 \) (Correct) 2. **Right-hand Limit:** \(\lim_{x \to 5^+} f(x)\) Evaluating using \(\frac{5}{x+5}\): \[ = \frac{5}{5+5} = \frac{5}{10} = 0.5 \] Note: The image incorrectly shows \(-30\) as the value, which is a mistake. The discrepancy between the left-hand limit and right-hand limit at \( x = 5 \) illustrates a jump discontinuity, confirming that \( f(x) \) is not continuous at this point.
Expert Solution
Step 1: Given function

f open parentheses x close parentheses equals open curly brackets table row cell 3 x minus 7 space i f end cell cell x less than 5 end cell row cell fraction numerator 5 over denominator x plus 5 end fraction space space i f end cell cell x greater or equal than 5 end cell end table close

steps

Step by step

Solved in 3 steps with 6 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS