Limits

The limit notation

The image attached on the side is a visual representation of how the limit notation is written. The lim stands for limit and is just a way for easier recognition of the notation. Just like the symbol for pi represents a certain number, or x usually representing the unknown value in an equation.

f(x) represents the function under consideration.

x-->a is usually reads "as x approaches a" and would mean that as x approaches a certain value, represented by a. c on the right side of the limit would be the value it closes in on. Hence we say that "limit of f(x), as x approaches a, equals c". It can also be read as "As f(x) approaches c, x approaches a". The way you read it is not important in itself, but it can be good to know about for future reference.

Another way to write it, which will be used here as google sites is not the best website for writing down maths ... anything, is:

f(x)-->c as x-->c

What is a limit?

Simply put, a limit of a function is when the value of a function gets closer, but not equal, to a certain value, as x gets closer to another number.

To show this, we have the function f(x) = x^2 - x +2. When x=1 (f(1)) the value of f(x) is 2. Continuing, f(1.5)=2.75, f(1.8)= 3.44, f(1.9)=3.71 and f(1.99)=3.97. All these values of x are under 2, but we can also do from values over 2. In that case, we get that f(3)=8, f(2.5)=5.75, f(2.2)=4.64, f(2.1)=4.31, f(2.01) = 4.03. As you can see, all of these values are close to 4, but not exactly 4. We can therefore say that f(x) gets closer to 4, as x gets closer to 2.

If we use the limit notation in this situation, we get:

f(x)-->4 as x--> 2

Here, I feel it's important to note that, in a limit, f(x) can get infinitely close to the limit, but can never be exactly equal to it. In this case, the function can get infinitely close to 4, but never be four.

This is especially relevant in piecewise functions where the values of the function can change drastically for just one number, then turn back. For the function stated above, we could say that at x=2 the function would turn into f(x) = 500x^2 + 50x +1000, which is a would be a very different value than that of the expected.

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