Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Limits
When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topic…
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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.
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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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