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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Question
![**Problem Statement:**
Find the following limit or state that it does not exist.
\[ \lim_{{x \to 2}} \frac{{x - 2}}{{\sqrt{2x + 5} - 3}} \]
**Explanation:**
To find the limit of the given function as \(x\) approaches 2, we need to analyze the behavior of the function near \(x = 2\). If direct substitution leads to an indeterminate form like 0/0, we may need to apply algebraic techniques such as rationalizing the denominator, simplifying, or using L'Hôpital's rule if appropriate.
**Solution Approach:**
1. Substitute \(x = 2\) directly to check for an indeterminate form.
- If it results in \(\frac{0}{0}\), apply an appropriate method to simplify the expression.
2. Rationalize the denominator or use algebraic simplification to resolve the indeterminate form.
3. Calculate the limit after simplification.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ad87420-52d2-4e34-815e-eb9bcb5c95c0%2F2ac5be10-8b52-409f-9495-e1da07301456%2Fzhddxg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the following limit or state that it does not exist.
\[ \lim_{{x \to 2}} \frac{{x - 2}}{{\sqrt{2x + 5} - 3}} \]
**Explanation:**
To find the limit of the given function as \(x\) approaches 2, we need to analyze the behavior of the function near \(x = 2\). If direct substitution leads to an indeterminate form like 0/0, we may need to apply algebraic techniques such as rationalizing the denominator, simplifying, or using L'Hôpital's rule if appropriate.
**Solution Approach:**
1. Substitute \(x = 2\) directly to check for an indeterminate form.
- If it results in \(\frac{0}{0}\), apply an appropriate method to simplify the expression.
2. Rationalize the denominator or use algebraic simplification to resolve the indeterminate form.
3. Calculate the limit after simplification.
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