Graphing and Visualizing Limits
It's super helpful to plug numbers into the function and see the output. It's even more helpful to graph the results. Try to draw or imagine how a function actually looks. Is it really hip to be x2? Draw the graph and decide for yourself.
Let y = f(x) = x3 – 2. As x gets close to zero, what does y approach?
As x approaches zero, y approaches -2. There are several different ways to say this:
- As x gets close to 0, y gets close to -2.
- As x gets close to 0, f(x) gets close to -2.
- As x approaches 0, y approaches -2.
- As x approaches 0, f(x) approaches -2.
- As x goes to 0, y goes to -2.
- As x goes to 0, f(x) goes to -2.
Each of these phrases mean the same thing. Here's yet another way to say it:
The limit of f(x) as x approaches 0 is -2.
We know what "x approaches 0" means. The limit of f(x) is the value f(x) is getting close to.
We can have x approach other numbers besides 0.
Let y = f(x) = cos(x). What is the limit of f(x) as x approaches 2π?
Moving x around, we see that as x gets closer to 2π, f(x) gets close to 1. The limit of f(x) as x approaches 2π is 1.
This is the basic idea behind limits. We look at what a function does as the independent variable, or input, gets closer and closer to some specified value.