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- Homework Statement
- ε is a positive infinitesimal and you need to determine whether the given expression is infinitesimal, finite but not infinitesimal, or infinite:

##\frac {\sqrt{4+ε} ~~-2} {ε}##

- Relevant Equations
- Looking at just the square root in the numerator for now.

I am thinking that 4+ε in the numerator is finite and therefore ## \sqrt{4+ε}## would also be finite. However I am unsure as a video seems to indicate this would be infinitesimal.

Hello All. This is my first post on the Physics Forums. I have started to self-study calculus and based on the feedback from this site and others, I have chosen Elementary Calculus: An Infinitesimal Approach by Jerome Keisler.

I am working through the problems for section 1.5 (page 34/35).

https://people.math.wisc.edu/~keisler/chapter_1b.pdf

I am stuck on question 29 and in the process found that I am not completely sure of what the the sum of a finite and infinitesimal is (i.e. is it finite but not infinitesimal, or is it infinitesimal).

Question 29:

ε is a positive infinitesimal and you need to determine whether the given expression is infinitesimal, finite but not infinitesimal, or infinite:

##\frac {\sqrt{4+ε} ~~-2} {ε}##

Based on the rules of infinitesimal, finite, and infinite numbers (page 30/31) it is my understanding that 4+ε in the numerator is finite and therefore ## \sqrt{4+ε}## would also be finite. However; I am confused as there is a video series on YouTube which follows Keisler's text and indicates that

## \sqrt{1+ε}## is infinitesimal: https://youtu.be/yw0-wnEuaHc (watch at 9:30 and again at 12:10). Which is correct, and why?

I am working through the problems for section 1.5 (page 34/35).

https://people.math.wisc.edu/~keisler/chapter_1b.pdf

I am stuck on question 29 and in the process found that I am not completely sure of what the the sum of a finite and infinitesimal is (i.e. is it finite but not infinitesimal, or is it infinitesimal).

Question 29:

ε is a positive infinitesimal and you need to determine whether the given expression is infinitesimal, finite but not infinitesimal, or infinite:

##\frac {\sqrt{4+ε} ~~-2} {ε}##

Based on the rules of infinitesimal, finite, and infinite numbers (page 30/31) it is my understanding that 4+ε in the numerator is finite and therefore ## \sqrt{4+ε}## would also be finite. However; I am confused as there is a video series on YouTube which follows Keisler's text and indicates that

## \sqrt{1+ε}## is infinitesimal: https://youtu.be/yw0-wnEuaHc (watch at 9:30 and again at 12:10). Which is correct, and why?

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