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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![**Mathematics: Understanding Limits Involving Integer Functions**
In this lesson, we will analyze the left-hand limit of a function involving the integer part (floor function). The problem at hand is:
\[ \lim_{{x \to 1^{-}}} \text{INT}(x + 4) = \]
### Step-by-Step Solution
1. **Expression Breakdown**:
- The function given is \(\text{INT}(x + 4)\).
- Here, "INT" denotes the floor function, which returns the greatest integer less than or equal to \(x + 4\).
2. **Left-Hand Limit**:
- The limit as \(x\) approaches \(1\) from the left (\(x \to 1^{-}\)) means we consider values of \(x\) slightly less than \(1\).
3. **Substitute and Simplify**:
- Let \(x\) approach \(1\) from the left.
- Then \(x = 1 - \epsilon\) where \(\epsilon\) is a very small positive number.
- Substituting in the expression \((x + 4)\), we get:
\[
1 - \epsilon + 4 = 5 - \epsilon
\]
4. **Apply the Floor Function**:
- The value of \(5 - \epsilon\) will always be slightly less than \(5\) since \(\epsilon\) is a very small positive number.
- Hence, the floor function of \((5 - \epsilon)\) is \(\text{INT}(5 - \epsilon) = 4\).
5. **Conclusion**:
- Therefore,
\[
\lim_{{x \to 1^{-}}} \text{INT}(x + 4) = 4
\]
The final answer in the blank provided should be filled with **4**.
### Graphical Overview (Optional for Enhanced Understanding)
- If we were to plot the function \(\text{INT}(x+4)\), we would see a step graph.
- As \(x\) increases towards \(1\) from the left, \(x + 4\) increases towards \(5\) but does not include \(5\) before hitting the integer part value of \(5\).
- The value of the function, therefore,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdf785451-152a-464f-9db5-033c0b71374a%2F4bcb2262-6e5e-4b74-bf22-e17d1d5664ab%2Fziwfsgd_processed.png&w=3840&q=75)
Transcribed Image Text:**Mathematics: Understanding Limits Involving Integer Functions**
In this lesson, we will analyze the left-hand limit of a function involving the integer part (floor function). The problem at hand is:
\[ \lim_{{x \to 1^{-}}} \text{INT}(x + 4) = \]
### Step-by-Step Solution
1. **Expression Breakdown**:
- The function given is \(\text{INT}(x + 4)\).
- Here, "INT" denotes the floor function, which returns the greatest integer less than or equal to \(x + 4\).
2. **Left-Hand Limit**:
- The limit as \(x\) approaches \(1\) from the left (\(x \to 1^{-}\)) means we consider values of \(x\) slightly less than \(1\).
3. **Substitute and Simplify**:
- Let \(x\) approach \(1\) from the left.
- Then \(x = 1 - \epsilon\) where \(\epsilon\) is a very small positive number.
- Substituting in the expression \((x + 4)\), we get:
\[
1 - \epsilon + 4 = 5 - \epsilon
\]
4. **Apply the Floor Function**:
- The value of \(5 - \epsilon\) will always be slightly less than \(5\) since \(\epsilon\) is a very small positive number.
- Hence, the floor function of \((5 - \epsilon)\) is \(\text{INT}(5 - \epsilon) = 4\).
5. **Conclusion**:
- Therefore,
\[
\lim_{{x \to 1^{-}}} \text{INT}(x + 4) = 4
\]
The final answer in the blank provided should be filled with **4**.
### Graphical Overview (Optional for Enhanced Understanding)
- If we were to plot the function \(\text{INT}(x+4)\), we would see a step graph.
- As \(x\) increases towards \(1\) from the left, \(x + 4\) increases towards \(5\) but does not include \(5\) before hitting the integer part value of \(5\).
- The value of the function, therefore,
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