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.
Sample Question
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.
Sample Problem
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.
Exercise 1
Let y = f(x) = sin(x). What is the limit of f(x) as x approaches 0?
Exercise 2
Let y = f(x) = sin(x). What is the limit of f(x) as x approaches 2π?
Exercise 3
Let y = f(x) = sin(x). What is the limit of f(x) as x approaches π / 2?
Exercise 4
Let y = f(x) = sin(x). What is the limit of f(x) as x approaches -π?
Exercise 5
Let f(x) = x2 -4x + 3. Graph f(x) and use the graph to find the limit of f(x) as x approaches 0.
Exercise 6
Let f(x) = x2 -4x + 3. Graph f(x) and use the graph to find the limit of f(x) as x approaches 3.
Exercise 7
Let f(x) = x2 -4x + 3. Graph f(x) and use the graph to find the limit of f(x) as x approaches 1.
Exercise 8
Although we say "the limit of f(x) as x approaches 2 is 5," we write
limx → 2 f(x) = 5.
Let f(x) = 2 – 2x. Find the limit, limx → 2 f(x).
Exercise 9
Let f(x) = 2 – 2x. Find the limit, limx → 0 f(x).
Exercise 10
Let f(x) = 2 - 2x. Find the limit, limx → 1 f(x).