The expression (sqrt(4+x)-2)/x, where x is an infinitesimal number on the hyperreal line, is finite. Initially, the user concluded it was infinite by substituting sqrt(4+x) with (2+y), where y is another infinitesimal. However, multiplying both the numerator and denominator by (sqrt(4+x)+2) simplifies the expression to 1/(sqrt(4+x)+2), which is finite. This conclusion aligns with the correct interpretation of the behavior of infinitesimals.
PREREQUISITESStudents studying calculus, mathematicians exploring non-standard analysis, and anyone interested in the properties of infinitesimals and hyperreal numbers.
If ##\sqrt{4+x}=2+y##, where y is infinitesimal, you expression becomes ##\frac{y}{x}##. The quotient of two infinitesimals can be anything: infinitesimal, finite but not infinitesimal, or infinite. Your conclusion is therefore premature (and wrong according to your book).GwtBc said:Homework Statement
Say x is an infinitesimal number on the hyperreal line, is this expression finite, infinite or infinitesimal
Homework Equations
(sqrt(4+x)-2)/x
The Attempt at a Solution
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My approach so far has been that sqrt(4+x) is (2+y) where y is another infinitesimal and y<x, which seems to lead to the conclusion that this expression is infinite, but the answers say it's finite. Any help?
Samy_A said:If ##\sqrt{4+x}=2+y##, where y is infinitesimal, you expression becomes ##\frac{y}{x}##. The quotient of two infinitesimals can be anything: infinitesimal, finite but not infinitesimal, or infinite. Your conclusion is therefore premature (and wrong according to your book).
What happens when you muliply both numerator and denominator by ##\sqrt{4+x}+2##?