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##?
An infinitesimal expression is one that approaches zero as the variable or value in the expression approaches a certain limit. A finite expression is one that has a definite value, not approaching zero or infinity. An infinite expression is one that has no definite value, growing without bound.
To determine the nature of an expression, you can analyze its behavior as the variable or value approaches certain limits. If the expression approaches a non-zero value, it is finite. If the expression approaches zero, it is infinitesimal. If the expression grows without bound, it is infinite.
Examples of infinitesimal expressions include 1/x as x approaches infinity, or sin(x)/x as x approaches 0. Finite expressions include 2x + 3, or 1/x as x approaches a finite value. Infinite expressions include 1/x as x approaches 0, or 2^x as x approaches infinity.
No, an expression cannot be both infinitesimal and infinite. These terms describe opposite behaviors of an expression as it approaches certain limits. If an expression approaches zero, it cannot also grow without bound.
Understanding the nature of an expression is important in many areas of science and mathematics. It allows for better analysis and prediction of the behavior of a system or function. It also helps in identifying limits and boundaries in solving problems and making calculations.