Unit 1 - Objective 1 - Limits/Continuity

The limit of a function is that value which the function approaches as x approaches the given value of a.

Definition: A function f(x) is continuous at x = a if all three of the following conditions are satisfied:

1. f(a) exists
2. exists
3. f(a) =

Given the following function f(x):

1. f(-3) exists and equals the y value which is -2
2. exists and equals the y value which is -2 because the function approaches -2 as you approach x = -3 from the left and right
3. because -2 = -2
   

So f(x) is continuous at x = -3

At the point x = -1:

1. f(-1) does not exist

So f(x) is not continuous at x = -1

At the point x = 1:

1. f(1) exists and equals the y value which is 0
2. does not exist because the function approaches 2 from the right and 0 from the left

In order for to exist you must approach the same number from the right and left. Since you get different values the does not exist.

So f(x) is not continuous at x = 1.

Evaluate each of the following limits

Problem 1

Problem 2

does not exist

Problem 3

Problem 4

Problem 5