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Introduction to Functions, Graphs, And Limits - At A Glance:

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) = x- 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.

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

limto 2 f(x) = 5.

Let f(x) = 2 - 2x. Find the limit.

  • limx to 2 f(x)

Exercise 9

Let f(x) = 2 - 2x. Find the limit. 

  • limx to 0 f(x)

Exercise 10

Let f(x) = 2 - 2x. Find the limit. 

  • limx to 1 f(x)
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