Fundamental Mathematics/Arithmetic/Limits

A limit looks at what happens to a function when the input approaches a certain value. The general notation for a limit is as follows:

${\displaystyle \lim _{x\to a}f(x)=L}$

This is read as "The limit of ${\displaystyle f}$ of ${\displaystyle x}$ as ${\displaystyle x}$ approaches ${\displaystyle a}$". We'll take up later the question of how we can determine whether a limit exists for ${\displaystyle f(x)}$ at ${\displaystyle a}$ and, if so, what it is. For now, we'll look at it from an intuitive standpoint.

Examples

Example 1

Given ${\displaystyle f(x)=x^{2}}$ , find ${\displaystyle x}$ approaches ${\displaystyle 2}$

Using the above notation, we can write the limit that we're interested in as follows:

${\displaystyle \lim _{x\to 2}x^{2}}$

One way to try to evaluate what this limit is would be to choose values near 2, compute ${\displaystyle f(x)}$ for each, and see what happens as they get closer to 2. This is implemented as follows:

 ${\displaystyle x}$ ${\displaystyle f(x)=x^{2}}$ 1.7 1.8 1.9 1.95 1.99 1.999 2.89 3.24 3.61 3.8025 3.9601 3.996

Here we chose numbers smaller than 2, and approached 2 from below. We can also choose numbers larger than 2, and approach 2 from above:

 ${\displaystyle x}$ ${\displaystyle f(x)=x^{2}}$ 2.3 2.2 2.1 2.05 2.01 2.001 5.29 4.84 4.41 4.2025 4.0401 4.004

We can see from the tables that as ${\displaystyle x}$ grows closer and closer to 2, ${\displaystyle f(x)}$ seems to get closer and closer to 4, regardless of whether ${\displaystyle x}$ approaches 2 from above or from below. For this reason, we feel reasonably confident that the limit of ${\displaystyle x^{2}}$ as ${\displaystyle x}$ approaches 2 is 4, or, written in limit notation,

${\displaystyle \lim _{x\to 2}x^{2}=4}$

Example 2

We could have also just substituted 2 into ${\displaystyle x^{2}}$ and evaluated: ${\displaystyle (2)^{2}=4}$ . However, this will not work with all limits.

Now let's look at another example. Suppose we're interested in the behavior of the function ${\displaystyle f(x)={\frac {1}{x-2}}}$ as ${\displaystyle x}$ approaches 2. Here's the limit in limit notation:

${\displaystyle \lim _{x\to 2}{\frac {1}{x-2}}}$

Just as before, we can compute function values as ${\displaystyle x}$ approaches 2 from below and from above. Here's a table, approaching from below:

 ${\displaystyle x}$ ${\displaystyle f(x)={\frac {1}{x-2}}}$ 1.7 1.8 1.9 1.95 1.99 1.999 -3.333 -5 -10 -20 -100 -1000

And here from above:

 ${\displaystyle x}$ ${\displaystyle f(x)={\frac {1}{x-2}}}$ 2.3 2.2 2.1 2.05 2.01 2.001 3.333 5 10 20 100 1000

In this case, the function doesn't seem to be approaching a single value as ${\displaystyle x}$ approaches 2, but instead becomes an extremely large positive or negative number (depending on the direction of approach). This is known as an infinite limit. Note that we cannot just substitute 2 into ${\displaystyle {\frac {1}{x-2}}}$ and evaluate as we could with the first example, since we would be dividing by 0.

Eample 3

${\displaystyle f(x)={\frac {x^{2}(x-2)}{x-2}}}$

This function is the same as

${\displaystyle f(x)=\left\{{\begin{matrix}x^{2}&{\text{if }}x\neq 2\\{\mbox{undefined}}&{\text{if }}x=2\end{matrix}}\right.}$

Note that these functions are really completely identical; not just "almost the same," but actually, in terms of the definition of a function, completely the same; they give exactly the same output for every input.

In elementary algebra, a typical approach is to simply say that we can cancel the term ${\displaystyle (x-2)}$ , and then we have the function ${\displaystyle f(x)=x^{2}}$ . However, that would be inaccurate; the function that we have now is not really the same as the one we started with, because it is defined when ${\displaystyle x=2}$ , and our original function was specifically not defined when ${\displaystyle x=2}$ . This may seem like a minor point, but from making this kind of assumptions we can easily derive absurd results, such that ${\displaystyle 0=1}$ (see Mathematical Fallacy § All numbers equal all other numbers in Wikipedia for a complete example). Even without calculus we can avoid this error by stating that:

${\displaystyle f(x)={\frac {x^{2}(x-2)}{x-2}}=x^{2}{\text{ when }}x\neq 2}$

In calculus, we can introduce a more intuitive and also correct way of looking at this type of function. What we want is to be able to say that, although the function isn't defined when ${\displaystyle x=2}$, it works almost as if it was. It may not get there, but it gets really, really close. For instance, ${\displaystyle f(1.99999)=3.99996}$. The only question that we have is: what do we mean by "close"?