Notation Question - Indicating Asymptotic Values

Question

How would you mathematically indicate a value "right next to" another value

For example, the value "just before 4" is $4-.0000...1$, right?

Or the theoretical minimum $Y$ value of y=2^x is $0+0,000...1$ (because of the asymptote) right?

Thanks.

Answer 1

You could use $4 + \epsilon$ or $4^+$ to symbolically represent real numbers infinitesimally close to but greater than $4$. For less than $4$, you could use $4 - \epsilon$ or $4^-$. But neither of these can represent the real number "right next to $4$", as mentioned in other answers.

Answer 2

A lot of times, if we want to express some value very close to $x$, we write $x+\epsilon$, for all $\epsilon>0$. This fits nicely in with a lot of the definitions of limits, and other calculus concepts.

Answer 3

In the real numbers there is no value “just before $4$”. This might be counterintuitive to you but maybe thinking about it this way helps: Imagine the number line and any point $x$ to the left of $4$. Because $x$ is not $4$ itself, there is a small distance between these points. Go half this distance and you reach another point ($\frac{1}{2} (x + 4)$, actually) that is still to the left of $4$ but closer to $4$ than $x$. Because we can do this for any such $x$, there can’t be one that is closest to $4$.

The fact that these numbers don’t exists is then one motivation to develop calculus in the way we do.