I'm going to Keisler's "Elementary Calculus, an Infinitesimal approach" , and I'm stuck on a problem:
Given that $H$ is a positive infinite term, determine whether the given expression is infinitesimal, finite (but not infinitesimal), or infinite:
$$H-\sqrt{H+1}\sqrt{H+2}$$
The issue I'm having is that the book introduced rules for operations between various types of terms (multiplication of an infinitesimal and a finite, for example), but also explained that certain operations, such as subtraction between infinite terms or multiplication between infinite and infinitesimals, are "indeterminate forms", where the answer depends on what those terms actually are.
So I'm not sure how to rearrange this expression to avoid such indeterminate form, though I've been able to do it for previous 35 problems.
Any ideas?
I am not sure what the "rules for operations" are & which computations Keisler wants to avoid.
Here is my way to look at it :
$$X=H-\sqrt{H+2}\sqrt{H+1}$$ $$X=H-\sqrt{(H+2)(H+1)}$$ $$X=H-\sqrt{H^2+3H+1}$$ $$X=H-\sqrt{H^2+2 \times H \times (3/2) +(3/2)^2-(3/2)^2+1}$$ $$X=H-\sqrt{(H+(3/2))^2-(3/2)^2+1}$$
When $H$ is very large :
$$X \approx H-(H+(3/2))$$ $$X \approx (-3/2)$$
It is Negative & finite.