Which of the following limits is equal to f1(x) (i.e. derivative of f), where f(x)=√x² +1 − 2x lim h→0 lim h→0 lim h→0 √h² +1-1-2h h lim h→0 √(x + h)² +1 − 2h h √(x + h)² + 1 - √x² +1 − 2h h √(x + h)² +1 − 2x h lim x→0 √x² + 1-2x ₂2 x
Which of the following limits is equal to f1(x) (i.e. derivative of f), where f(x)=√x² +1 − 2x lim h→0 lim h→0 lim h→0 √h² +1-1-2h h lim h→0 √(x + h)² +1 − 2h h √(x + h)² + 1 - √x² +1 − 2h h √(x + h)² +1 − 2x h lim x→0 √x² + 1-2x ₂2 x
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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![### Derivatives and Limits
#### Question:
Which of the following limits is equal to \( f'(x) \) (i.e., the derivative of \( f \)), where
\[ f(x) = \sqrt{x^2 + 1 - 2x} \]
##### Options:
1.
\[
\lim_{{h \to 0}} \frac{\sqrt{h^2 + 1 - 1 - 2h}}{h}
\]
2.
\[
\lim_{{h \to 0}} \frac{\sqrt{(x + h)^2 + 1 - 2h}}{h}
\]
3.
\[
\lim_{{h \to 0}} \frac{\sqrt{(x + h)^2 + 1 - \sqrt{x^2 + 1 - 2h}}}{h}
\]
4.
\[
\lim_{{h \to 0}} \frac{\sqrt{(x + h)^2 + 1 - 2x}}{h}
\]
5.
\[
\lim_{{x \to 0}} \frac{\sqrt{x^2 + 1 - 2x}}{x}
\]
#### Explanation:
To find the correct derivative expression, we need to identify the limit definition of the derivative:
\[
f'(x) = \lim_{{h \to 0}} \frac{f(x + h) - f(x)}{h}
\]
Given that:
\[ f(x) = \sqrt{x^2 + 1 - 2x},\]
we substitute \( f(x) \) into the limit definition to see which option matches.
#### Detailed Solution:
1. The first option should not contain any terms unrelated to \( f(x) \).
2. The second option lacks \( f(x) \) in the numerator, making it incorrect.
3. The third option adds unnecessary terms.
4. The fourth option matches the structure of the limit definition of the derivative.
5. The fifth option evaluates the limit as \( x \to 0 \), which is not correct for the derivative at an arbitrary \( x \).
Thus, the correct answer must be:
\[
\lim_{{h \to 0}} \frac{\sqrt{(x + h)^2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4c3310c5-cf1b-4860-895e-b0d1f5af2b55%2F2bf69a39-89f7-46ae-8d31-78ef082b531c%2Feiy0yae_processed.png&w=3840&q=75)
Transcribed Image Text:### Derivatives and Limits
#### Question:
Which of the following limits is equal to \( f'(x) \) (i.e., the derivative of \( f \)), where
\[ f(x) = \sqrt{x^2 + 1 - 2x} \]
##### Options:
1.
\[
\lim_{{h \to 0}} \frac{\sqrt{h^2 + 1 - 1 - 2h}}{h}
\]
2.
\[
\lim_{{h \to 0}} \frac{\sqrt{(x + h)^2 + 1 - 2h}}{h}
\]
3.
\[
\lim_{{h \to 0}} \frac{\sqrt{(x + h)^2 + 1 - \sqrt{x^2 + 1 - 2h}}}{h}
\]
4.
\[
\lim_{{h \to 0}} \frac{\sqrt{(x + h)^2 + 1 - 2x}}{h}
\]
5.
\[
\lim_{{x \to 0}} \frac{\sqrt{x^2 + 1 - 2x}}{x}
\]
#### Explanation:
To find the correct derivative expression, we need to identify the limit definition of the derivative:
\[
f'(x) = \lim_{{h \to 0}} \frac{f(x + h) - f(x)}{h}
\]
Given that:
\[ f(x) = \sqrt{x^2 + 1 - 2x},\]
we substitute \( f(x) \) into the limit definition to see which option matches.
#### Detailed Solution:
1. The first option should not contain any terms unrelated to \( f(x) \).
2. The second option lacks \( f(x) \) in the numerator, making it incorrect.
3. The third option adds unnecessary terms.
4. The fourth option matches the structure of the limit definition of the derivative.
5. The fifth option evaluates the limit as \( x \to 0 \), which is not correct for the derivative at an arbitrary \( x \).
Thus, the correct answer must be:
\[
\lim_{{h \to 0}} \frac{\sqrt{(x + h)^2
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