Difference between $\lim_{x \to 0} \text{ and } x=0$

Question

My question is about infinity and $0$. Using the same logic as $0.999...$ is equal to $1$, if a number is depreciating towards $0$ infinitely at a constant rate (say it is being constantly divided by two), would it be correct to say that it would eventually reach $0$. The graph would have an asymptote of $0$, because there are an infinite amount of numbers less than $1$. But if there are an infinite amount of $0$s after the decimal point, even if they are eventually followed by a $1$, would it equal $0$? It is hard to comprehend because if the $0$s go on infinitely, the $1$ would never actually exist or have a value.

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I guess my question is: Would it ever get to a point where a number approaching $0$ would actually be considered $0$? I feel like it shouldn't since there would always be at least one digit that isn't a $0$, but it also feels like if there are an infinite number of $0$s, then the number is equal to $0$.

Answer 1

You mention an example: repeatedly dividing by 2. If you start with a number $c \neq 0$ and divide it by $2$ exactly $n$ times, you are left with the number $c / 2^n$. Even if you perform a large number of divisions (e.g., $n=1000$ or $n=$ Graham's number), $c / 2^n$ will always be nonzero.

Answer 2

No.You can't say $lim_{x \to 0}$ means the value of $x$ is simply $0$.If there is a condition like this, then it means that the domain of the function may be everything but obviously except $0$, Like,there is a situation $$\lim_{x \to 0}\dfrac{ax+b}{x}$$ So, here $x$ is approaching to $0$(may be as close as possible to $0$).But is undefined when it IS equals to $0$.So,you can't say that.

Answer 3

(1).What is a number? The definition of the structure called the "real" number system ($\Bbb R$) implies that $\Bbb R$ has no positive members that are less than all positive rationals. This is not trivial, because we can extend $\Bbb R$ to a larger arithmetic structure that does have such "infiniteimals".

(2). The numerical meaning, in $\Bbb R,$ of an infinite sequence $0.a_1a_2a_3...$ of decimal-place values is the least $r\in \Bbb R$ such that $r \ge a_1/10+...+a_n/10^n$ for each $n\in \Bbb N.$ The existence of the $least$ such $r$ also comes from the def'n of $\Bbb R$.

(3). What is an infinite sequence? It is a function $f$ whose domain is $\Bbb N.$ But we often write terms like $a_n$ instead of $f(n).$ An infinite sequence of digits is cannot followed by some other digit of the same sequence because the nth digit is $a_n=f(n)$ and there is no $m\in \Bbb N$ that comes after all the $n\in \Bbb N.$

I suggest you find a text that covers the definition of $\Bbb R$ in detail