College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter2: Functions
Section: Chapter Questions
Problem 30P: In this problem you are asked to find a function that models in real life situation and then use the...
Related questions
Question
Prameter k
![---
### Evaluating Limits with Piecewise Functions
#### Given Function:
\[
f(x) =
\begin{cases}
2k - x & \text{if } x \geq 1 \\
\dfrac{1}{k} + x & \text{if } x < 1
\end{cases}
\]
Where \( k \) is a parameter.
#### Problem Statement:
Evaluate \(\lim_{{x \to 1^-}} f(x)\)
#### Question:
Can you choose a proper value for \( k \) so that \(\lim_{{x \to 1}} f(x)\) exists?
---
In this example, you are presented with a piecewise function \( f(x) \) that is defined differently for values of \( x \) greater than or equal to 1 and values of \( x \) less than 1. Your task is to evaluate the left-hand limit (\( \lim_{{x \to 1^-}} \)) of the function and determine if there is a value of the parameter \( k \) that makes the overall limit as \( x \) approaches 1 exist.
Let's break down the function:
- For \( x \geq 1 \), the function is \( f(x) = 2k - x \).
- For \( x < 1 \), the function is \( f(x) = \frac{1}{k} + x \).
To find if the proper \( k \) makes \(\lim_{{x \to 1}} f(x)\) exist, we need to evaluate the limits from both sides and see if they are equal:
1. **Evaluate the left-hand limit (\( \lim_{{x \to 1^-}} \)):**
\[ \lim_{{x \to 1^-}} f(x) = \lim_{{x \to 1^-}} \left( \frac{1}{k} + x \right) = \frac{1}{k} + 1 \]
2. **Evaluate the right-hand limit (\( \lim_{{x \to 1^+}} \)):**
\[ \lim_{{x \to 1}} f(x) = \lim_{{x \to 1^+}} \left( 2k - x \right) = 2k - 1 \]
3.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fedc716be-e066-4234-bd1a-1a86837575c3%2F6bb53984-2bb8-47a9-be91-650f23967c36%2Fkxeh0tb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
### Evaluating Limits with Piecewise Functions
#### Given Function:
\[
f(x) =
\begin{cases}
2k - x & \text{if } x \geq 1 \\
\dfrac{1}{k} + x & \text{if } x < 1
\end{cases}
\]
Where \( k \) is a parameter.
#### Problem Statement:
Evaluate \(\lim_{{x \to 1^-}} f(x)\)
#### Question:
Can you choose a proper value for \( k \) so that \(\lim_{{x \to 1}} f(x)\) exists?
---
In this example, you are presented with a piecewise function \( f(x) \) that is defined differently for values of \( x \) greater than or equal to 1 and values of \( x \) less than 1. Your task is to evaluate the left-hand limit (\( \lim_{{x \to 1^-}} \)) of the function and determine if there is a value of the parameter \( k \) that makes the overall limit as \( x \) approaches 1 exist.
Let's break down the function:
- For \( x \geq 1 \), the function is \( f(x) = 2k - x \).
- For \( x < 1 \), the function is \( f(x) = \frac{1}{k} + x \).
To find if the proper \( k \) makes \(\lim_{{x \to 1}} f(x)\) exist, we need to evaluate the limits from both sides and see if they are equal:
1. **Evaluate the left-hand limit (\( \lim_{{x \to 1^-}} \)):**
\[ \lim_{{x \to 1^-}} f(x) = \lim_{{x \to 1^-}} \left( \frac{1}{k} + x \right) = \frac{1}{k} + 1 \]
2. **Evaluate the right-hand limit (\( \lim_{{x \to 1^+}} \)):**
\[ \lim_{{x \to 1}} f(x) = \lim_{{x \to 1^+}} \left( 2k - x \right) = 2k - 1 \]
3.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
Similar questions
Recommended textbooks for you
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Algebra for College Students
Algebra
ISBN:
9781285195780
Author:
Jerome E. Kaufmann, Karen L. Schwitters
Publisher:
Cengage Learning
Intermediate Algebra
Algebra
ISBN:
9781285195728
Author:
Jerome E. Kaufmann, Karen L. Schwitters
Publisher:
Cengage Learning
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Algebra for College Students
Algebra
ISBN:
9781285195780
Author:
Jerome E. Kaufmann, Karen L. Schwitters
Publisher:
Cengage Learning
Intermediate Algebra
Algebra
ISBN:
9781285195728
Author:
Jerome E. Kaufmann, Karen L. Schwitters
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Algebra: Structure And Method, Book 1
Algebra
ISBN:
9780395977224
Author:
Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:
McDougal Littell