"Definition" of limit by plugging in numbers¶
On this page, the word "function" means a function that takes in a real number as its only argument, and evaluates to another real number. Some real numbers might not be valid inputs of the function; for example, square root is a function that accepts only nonnegative inputs.
Limit is a very important concept in calculus. Basically, all calculus is based on limits. The definition of derivative contains a limit. The definition of integral doesn't directly use limits, but integrals are almost always calculated using derivatives.
There are two common ways to introduce limits: teaching students to calculate limits without talking much about why they work like they do (high schools), and proving all the properties of limits needed very carefully with somewhat difficult proofs (university).
The "university" approach is not easy, so we start with the "high school" approach, and then move on to a definition that resembles the "university" approach a lot but is hopefully easier. On this page, we explore the "high school" approach and discuss its advantages and limitations.
Let's get started by plugging some numbers into the function
undefined | |
The values of
We notice a similar thing if we plug in values that are just a little bit less than
In this situation, we say that the limit of
To write a limit, you don't need to give a name to the function (
Simple examples¶
We have the following limit:
Here is an even simpler example:
Of course, these also work with any other numbers instead of 3 and 5.
Surprising example¶
Let
For example,
Similarly, to calculate
Let's try to calculate
The function is outputting zero, so it seems that the limit is zero. Let's also try negative numbers:
We still get zero, so it is tempting to conclude that the limit is zero,
but actually, the limit is not zero.
To see why, consider the following (carefully chosen) sequence of small numbers.
(To get the next
Now it seems like the limit should be 1.
The behavior of this function seems confusing. To get a better idea of it, let's draw a graph of it:
The function oscillates between
In this situation, we say that the limit doesn't exist or is undefined.
A limit might not exist.
Problems¶
So far we have calculated limits by plugging in nearby numbers. While this works, this method also has downsides:
- In complicated cases, you might not notice some behavior of the function. See the above example.
- There's no good way to plug in nearby numbers
if the limit contains unknown variables,
e.g.
. Solving unknown things later is a very powerful technique in math, so it's important that it works. - It's hard to prove anything about limits this way. "Just plug in nearby numbers" is not a good definition, so if you try to prove something with it, the "proof" won't be convincing.
To fix these problems, next we define limits precisely.