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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![### Calculus Problem: Finding the Limit
#### Problem Statement:
**Find the limit.**
4) Determine the limit.
\[ \lim_{{x \to -10^ -}} f(x), \text{ where } f(x) = \frac{1}{x + 10} \]
#### Explanation:
To solve this limit problem, you need to determine the behavior of the function \( f(x) = \frac{1}{x + 10} \) as \( x \) approaches -10 from the left-hand side (denoted by \( -10^- \)).
For an in-depth analysis:
1. **Identify the function and the approach direction:**
- The function given is \( f(x) = \frac{1}{x + 10} \).
- We're interested in the behavior of \( f(x) \) as \( x \) gets close to -10 from values less than -10.
2. **Analyze the function around the point of interest:**
- As \( x \to -10^- \), the value \( x + 10 \to 0^- \) because \( x \) is slightly less than -10.
- Consequently, \( \frac{1}{x + 10} \to -\infty \) as the denominator approaches zero from the negative side.
Therefore, the limit is:
\[ \lim_{{x \to -10^-}} \frac{1}{x + 10} = -\infty \]
This result indicates that as \( x \) approaches -10 from the left, the function value decreases without bound to negative infinity.
#### Notes:
- **Left-Hand Limit (LHL):** The notation \( -10^- \) signifies the limit is being taken from the left side (i.e., values approaching -10 from the negative side).
- **Behavior Near Asymptotes:** This type of limit often occurs with functions that have vertical asymptotes, in this case at \( x = -10 \).
This concludes the solution for determining the limit as \( x \) approaches -10 from the left for the given function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1a0a3e2b-c166-4815-91b8-41dfa54aca6a%2F8e97a8cb-e0ff-4522-9c8d-ecc0526dc33e%2Fm3ymm3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculus Problem: Finding the Limit
#### Problem Statement:
**Find the limit.**
4) Determine the limit.
\[ \lim_{{x \to -10^ -}} f(x), \text{ where } f(x) = \frac{1}{x + 10} \]
#### Explanation:
To solve this limit problem, you need to determine the behavior of the function \( f(x) = \frac{1}{x + 10} \) as \( x \) approaches -10 from the left-hand side (denoted by \( -10^- \)).
For an in-depth analysis:
1. **Identify the function and the approach direction:**
- The function given is \( f(x) = \frac{1}{x + 10} \).
- We're interested in the behavior of \( f(x) \) as \( x \) gets close to -10 from values less than -10.
2. **Analyze the function around the point of interest:**
- As \( x \to -10^- \), the value \( x + 10 \to 0^- \) because \( x \) is slightly less than -10.
- Consequently, \( \frac{1}{x + 10} \to -\infty \) as the denominator approaches zero from the negative side.
Therefore, the limit is:
\[ \lim_{{x \to -10^-}} \frac{1}{x + 10} = -\infty \]
This result indicates that as \( x \) approaches -10 from the left, the function value decreases without bound to negative infinity.
#### Notes:
- **Left-Hand Limit (LHL):** The notation \( -10^- \) signifies the limit is being taken from the left side (i.e., values approaching -10 from the negative side).
- **Behavior Near Asymptotes:** This type of limit often occurs with functions that have vertical asymptotes, in this case at \( x = -10 \).
This concludes the solution for determining the limit as \( x \) approaches -10 from the left for the given function.
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