Explaining A Mathematical Statement In Limitation Notation To A Layman

Question

I've never learnt limit notation or calculus before. However, during asking a question about what exactly a horizontal asymptote is, the definition I got was "We say that the line $y=k$ is an horizontal asymptote of the graph of $f$ when (and only when) $\lim_{x\to+\infty} f(x)=k$" , and that "the assertion $\lim_{x\to+\infty} f(x)=k$ means that for every number $\epsilon>0$, there is a number $M \in \mathbb{R}^+$ such that $x>M \implies |f(x)−k|<\epsilon$." I have two question about this:

1. What does this statement in limit notation above mean in layman's terms?
2. And also, from this statement, what does the horizontal asymptote mean in a layman's terms?

Answer 1

I like to think about limits to infinity this way: No matter what $\varepsilon$ you choose (as long as it's not the limit $k$ of the function), you can always find a large enough $x$ (denoted $x > M$), so that $f(x)$ will always lie between $k$ and $k\pm \varepsilon$ for all $x>M$. Thus, as you choose $\varepsilon$ to be arbitrarily small, $f(x)$ gets arbitrarily close to $k$.

(Obviously this isn't a rigorous statement, just an intuitive look at them).

In the case of $f(x)$ having a horizontal asymptote at $y=k$, $f(x)-k$ will get closer and closer to $0$ by taking a large enough value of $x$.

Answer 2

In layman terms $\lim\limits_{x \rightarrow \infty} f(x) = k$ (which is just a notation for your second statement that defines what this notation means) says that the bigger values you input to the function $f$, the closer it's output value is to $k$. In fact to be more precise it says that your function's values can get arbitrary close to $k$ as you like provided that you put in large enough input.

If somebody gives you a small positive number $\epsilon$, the statement says that you can find a number $M$, such that the part of the graph of your function that's from $M$ rightward on the $x$ axis will within a distance of $\epsilon$ from the horizontal line $k$.

Answer 3

Symbolically, a limit to infinity $\lim_{x \to \infty} f(x) = L$ can be interpreted as

$$ \forall \varepsilon > 0, \exists M, \forall x > M, |f(x)-L| \leq \varepsilon $$

There's an appealing (to me) game-theoretical interpretation that is part of the foundation of independence-friendly logic, in which the semantics of such a statement can be understood as a kind of game between two players, the Falsifier (or Abelard), symbolized by $\forall$, and the Verifier (or Eloise), symbolized by $\exists$. I'll explain this game assuming that there are no negations; negations complicate things a little, but there are ways to deal with them.

The players take turns picking values for the innermost assertion (in this case, $|f(x)-L| \leq \varepsilon$), with the Falsifier picking any value that is universally quantified, and the Verifier picking any value that is existentially quantified. As their names imply, the Falsifier is trying to pick values that will invalidate the innermost assertion, while the Verifier is trying to pick those that will sustain that assertion. We might imagine the following conversation:

Falsifier. I bet you can't get within $\varepsilon = 1/10$ of the limit $L$.

Verifier. I bet I can, provided I set $M = 100$; I bet you can't find an $x > 100$ such that $f(x)$ doesn't fall within $\varepsilon = 1/10$ of $L$.

[At this point, the Falsifier either fails or succeeds in selecting such an $x$.]

If both players play optimally, then the overall assertion is true or false depending on whether the innermost assertion is invalidated or sustained.