Definition of Limit¶

So far, we have "defined" limits by plugging in nearby numbers. There were several problems with this approach. On this page we fix them by defining limits precisely.

Close enough to¶

Before defining limits, we define what "close enough to" means in this context. This concept will simplify the definition of limit a lot.

An open interval is denoted with $(a,b)$, where $a < b$, and it means all numbers strictly between $a$ and $b$. For example, "$x$ is in the interval $(3,4)$" means same as $3 < x < 4$.

We say that an open interval is centered around $a$ if it's $(a-r, a+r)$ with some $r > 0$. Here $r$ is sometimes called the radius of the interval. For example, $(3,4)$ is centered around $3.5$ with radius $0.5$.

The saying "$x$ is close enough to $a$" means same as "$x$ is in some open interval centered around $a$ and $x \ne a$".

Examples:

• We have $\frac{x^3}{x} < 1$ on the interval $(-1, 1)$ (marked in green), except at its center $1$ where the function divides by zero. Therefore we can say that $\frac{x^3}{x} < 1$ when $x$ is close enough to $0$.

  

• We also have $\frac{x^3}{x} < 0.1$ when $x$ is close enough to $0$. (We could calculate what exactly the interval centered around $0$ is, but it's not important.)

  

• We have $0 < x^2 < 0.1$ when $x$ is close enough to zero. The $0 < x^2$ part is false when $x=0$, because then it says $0 < 0$. However, this doesn't matter, because the center $0$ is ignored in the definition of "close enough to".

  

• We have $x^2 < 0.1$ when $x$ is close enough to zero. This also happens to work when $x=0$, but it doesn't matter, because $x=0$ is ignored anyway.

• We do not have $x^2 < 4$ when $x$ is close enough to 2. This is because $x^2 > 4$ whenever $x > 2$, and any interval centered around 2 will contain numbers greater than 2.

  

• We have $x^2 < 5$ when $x$ is close enough to 2. For example, the interval $(1.8, 2.2)$ works, because the values will be smaller than $(2.2)^2 = 4.84$.

  

$\approx$ with tolerance¶

Let $a$ and $b$ be any numbers, and let $t$ be a positive number. We say that $a \approx b$ with tolerance $t$, if $b-t < a < b+t$. For example, $x \approx 3$ with tolerance $0.1$ means that $2.9 < x < 3.1$.

This is similar to the "close enough to" concept, but not quite the same. Specifically, "$x$ is close enough to $a$" means that $x \approx a$ with some tolerance, but also requires $x \ne a$.

The Definition¶

On the previous page, we noticed that $$ \lim_{x \to 1} \frac{x^3-1}{x-1} = 3 $$ by plugging in numbers like this:

The resulting values are approximately 3, but let's be more specific by specifying how close to 3 the values are.

Limit means that we can choose any tolerance we want, no matter how small. For example, we have $\frac{x^3-1}{x-1} \approx 3$ with tolerance $0.01$ (that is, $2.99 < \frac{x^3-1}{x-1} < 3.01$), if $x$ is close enough to 1. The point is that we can use any tolerance we want, such as $0.00000000001$.

Because we defined precisely what "$\approx$ with tolerance" and "close enough to" mean, there's nothing vague in this definition, so it's possible to write convincing proofs and derivations based on it.

Simple example: limit of $x$¶

We prove that $\lim_{x \to 3} x = 3$.

Let $t > 0$ be any tolerance. We want to show that $x \approx 3$ with tolerance $t$ when $x$ is close enough to $3$.

This is quite easy. Consider the interval $(3-t, 3+t)$. If $x$ is close enough to $3$, it is in $(3-t, 3+t)$, and $x \ne 3$. This means that $3-t < x < 3+t$, so we have $x \approx 3$ with tolerance $t$. (We didn't need $x \ne 3$ for anything.)

Of course, there's nothing special about the number 3. The same works with any other number.

Even simpler example: limit of a constant¶

We have $\lim_{x \to 3} 5 = 5$, because $5 \approx 5$ with any tolerance, regardless of what $x$ is.