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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Please help me find the limit and show work
![The image shows a mathematical limit problem:
\[
\lim_{{x \to 5}} \frac{{x - 1 - \sqrt{3x + 1}}}{{x - 5}}
\]
This expression represents the limit of a function as \( x \) approaches 5. The function is a rational expression with a numerator consisting of a subtraction between \( x - 1 \) and the square root of \( 3x + 1 \), and a denominator of \( x - 5 \). This type of problem typically requires techniques such as factoring, rationalizing, or applying L'Hopital's Rule to evaluate the limit, especially if direct substitution results in an indeterminate form like \(\frac{0}{0}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feddd6036-05ca-49ad-a1c3-fdec7d647876%2F65c3916c-8d91-4987-aa89-8d241f0659bb%2F0p1ea_processed.png&w=3840&q=75)
Transcribed Image Text:The image shows a mathematical limit problem:
\[
\lim_{{x \to 5}} \frac{{x - 1 - \sqrt{3x + 1}}}{{x - 5}}
\]
This expression represents the limit of a function as \( x \) approaches 5. The function is a rational expression with a numerator consisting of a subtraction between \( x - 1 \) and the square root of \( 3x + 1 \), and a denominator of \( x - 5 \). This type of problem typically requires techniques such as factoring, rationalizing, or applying L'Hopital's Rule to evaluate the limit, especially if direct substitution results in an indeterminate form like \(\frac{0}{0}\).
![The image displays a mathematical limit expression:
\[
\lim_{{x \to 3}} \frac{{x + 7}}{{(x - 3)^2}}
\]
This expression represents the limit of the function \(\frac{{x + 7}}{{(x - 3)^2}}\) as \(x\) approaches 3.
### Explanation:
- **Numerator:** \(x + 7\)
- **Denominator:** \((x - 3)^2\)
As \(x\) gets closer to 3, the denominator \((x - 3)^2\) approaches 0, which may cause the function to approach infinity or negative infinity, depending on the direction from which \(x\) approaches 3. This is an example of an indeterminate form.
### Graph:
While a graph is not present, typically a graph of this function would show a vertical asymptote at \(x = 3\) due to the denominator approaching 0. The behavior near the asymptote would need to be analyzed further to determine the exact nature of the limit.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feddd6036-05ca-49ad-a1c3-fdec7d647876%2F65c3916c-8d91-4987-aa89-8d241f0659bb%2Fs5dwqdg_processed.png&w=3840&q=75)
Transcribed Image Text:The image displays a mathematical limit expression:
\[
\lim_{{x \to 3}} \frac{{x + 7}}{{(x - 3)^2}}
\]
This expression represents the limit of the function \(\frac{{x + 7}}{{(x - 3)^2}}\) as \(x\) approaches 3.
### Explanation:
- **Numerator:** \(x + 7\)
- **Denominator:** \((x - 3)^2\)
As \(x\) gets closer to 3, the denominator \((x - 3)^2\) approaches 0, which may cause the function to approach infinity or negative infinity, depending on the direction from which \(x\) approaches 3. This is an example of an indeterminate form.
### Graph:
While a graph is not present, typically a graph of this function would show a vertical asymptote at \(x = 3\) due to the denominator approaching 0. The behavior near the asymptote would need to be analyzed further to determine the exact nature of the limit.
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