The notion of limit

Functions: Introduction to functions

The notion of limit

Limit

Let $a$ and $b$ be real numbers and let $f$ be a real function that is defined on an open interval containing $a$ .

We say that $f$ has limit $b$ at $a$ if $f(x)$ comes closer to $b$ as $x$ comes closer to $a$ .

In this case, we write $\textstyle\lim_{x\to a} f(x) = b$ or $\displaystyle\lim_{x\to a} f(x) = b$ .

To be more precise: $b$ is the limit of $f$ at $a$ if, for every (arbitrarily small) positive number $\epsilon$ a positive number $\delta$ can be found with $|b-f(x)|\le\epsilon$ for every $x\in\ivoo{a-\delta}{a+\delta}$ with $x\ne a$ .

If $b=\infty$ , then $b$ is the limit of $f$ at $a$ if there is a real number $c$ such that for every (arbitrarily large) number $M$ a positive number $\delta\lt |c-a|$ can be found with $f(x)\ge M$ for every $x\in\ivoo{a-\delta}{a+\delta}$ with $x\ne a$ .

If $b=-\infty$ , then $b$ is the limit of $f$ at $a$ if there is a real number $c$ such that for every (arbitrarily negative) number $M$ a positive number $\delta$ can be found with $f(x)\le M$ for every $x\in\ivoo{a-\delta}{a+\delta}$ with $x\ne a$ .

The rational function $f(x) = \dfrac{x-4}{x-4}$ is defined everywhere except at $4$ .

What is the limit of $f$ at $4$ ?

$\lim_{x\to 4}f(x)=$ $1$

This follows from the fact that $f(x)=1$ for every value of $x$ close to (but distinct from) $4$ .

New example