52. Let X if x < 1 if x = 1 3 g(x) = 2 - x? if 1 2 (a) Evaluate each of the following, if it exists. (i) lim g(x) (ii) lim g(x) x→1 (iii) g(1) x→1-
52. Let X if x < 1 if x = 1 3 g(x) = 2 - x? if 1 2 (a) Evaluate each of the following, if it exists. (i) lim g(x) (ii) lim g(x) x→1 (iii) g(1) x→1-
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...
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![**Problem 52.**
Let
\[
g(x) = \begin{cases}
x & \text{if } x < 1 \\
3 & \text{if } x = 1 \\
2 - x^2 & \text{if } 1 < x \leq 2 \\
x - 3 & \text{if } x > 2
\end{cases}
\]
(a) Evaluate each of the following, if it exists.
(i) \(\lim\limits_{x \to 1^-} g(x)\)
(ii) \(\lim\limits_{x \to 1} g(x)\)
(iii) \(g(1)\)
---
**Explanation:**
This problem involves evaluating the piecewise function \( g(x) \) at different points. The piecewise function is defined with four cases based on the values of \( x \):
1. For \( x \) less than 1, \( g(x) = x \).
2. For \( x \) equal to 1, \( g(x) = 3 \).
3. For \( x \) between 1 (not inclusive) and 2 (inclusive), \( g(x) = 2 - x^2 \).
4. For \( x \) greater than 2, \( g(x) = x - 3 \).
The goal is to evaluate the limits and the function value at \( x = 1 \).
(i) \(\lim\limits_{x \to 1^-} g(x)\) evaluates the limit of \( g(x) \) as \( x \) approaches 1 from the left (values less than 1).
(ii) \(\lim\limits_{x \to 1} g(x)\) evaluates the limit of \( g(x) \) as \( x \) approaches 1 from both sides (left and right).
(iii) \( g(1) \) evaluates the value of the function \( g(x) \) at \( x = 1 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0a4a1727-156d-4ade-b12f-025bad03a8ba%2F5667f7a8-b125-4374-a15b-a20f1de22787%2Fxb2tszp_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 52.**
Let
\[
g(x) = \begin{cases}
x & \text{if } x < 1 \\
3 & \text{if } x = 1 \\
2 - x^2 & \text{if } 1 < x \leq 2 \\
x - 3 & \text{if } x > 2
\end{cases}
\]
(a) Evaluate each of the following, if it exists.
(i) \(\lim\limits_{x \to 1^-} g(x)\)
(ii) \(\lim\limits_{x \to 1} g(x)\)
(iii) \(g(1)\)
---
**Explanation:**
This problem involves evaluating the piecewise function \( g(x) \) at different points. The piecewise function is defined with four cases based on the values of \( x \):
1. For \( x \) less than 1, \( g(x) = x \).
2. For \( x \) equal to 1, \( g(x) = 3 \).
3. For \( x \) between 1 (not inclusive) and 2 (inclusive), \( g(x) = 2 - x^2 \).
4. For \( x \) greater than 2, \( g(x) = x - 3 \).
The goal is to evaluate the limits and the function value at \( x = 1 \).
(i) \(\lim\limits_{x \to 1^-} g(x)\) evaluates the limit of \( g(x) \) as \( x \) approaches 1 from the left (values less than 1).
(ii) \(\lim\limits_{x \to 1} g(x)\) evaluates the limit of \( g(x) \) as \( x \) approaches 1 from both sides (left and right).
(iii) \( g(1) \) evaluates the value of the function \( g(x) \) at \( x = 1 \).
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