Need Help Determining Continuity of Functions

1. Jan 22, 2012

CJ256

1. The problem statement, all variables and given/known data

2. Relevant equations

Ability to graph functions.
Essential Discontinuity:
Jump discontinuity or Infinite discontinuity

3. The attempt at a solution

First Question:

After plugging in 2 for every equation and getting a result that was greater than 0, I determined that the function was continuous and that the type of discontinuity is Essential (Infinite) discontinuity. The reason why I chose infinite is because when I drew the graph on my TI-84 Plus it didn't seem to have an empty point and all my points that I tested were filled. I don't understand how graph them by hand. I usually have no problem solving this type of questions when I have a graph, but when I have to make my own I really struggle.

Second Question:

After testing a couple of values (I tested, -1, 0, 1 as possible values of A) I determined that the answer to the question is all negative values could be values of A. I really struggled with this because I did not understand what the question really asked me. I kind of tried to satisfy the equations and once I saw that it did I decided that that was the answer. I know this question is wrong so any help would be really appreciated.

2. Jan 22, 2012

SammyS

Staff Emeritus
Do you have a definition for a function being continuous a point ?

If so, what is that definition?

3. Jan 23, 2012

CJ256

Yep. A function f(x) is continuous at x = c if, as x approaches c as a limit, f(x) approaches f(c) as a limit or in other words this:

4. Jan 23, 2012

SammyS

Staff Emeritus
So, first of all $\displaystyle \lim_{x\,\to\,c}\,f(x)$ must exist. If it exists, then it must be equal to f(c).

First Problem: What is $\displaystyle \lim_{x\,\to\,2}\,f(x)\,?$

How do you determine whether or not this limit exists?

Second Problem: $\displaystyle \lim_{x\,\to\,3}\,f(x)\,?$

How do make sure this limit exists?

How do make sure this limit is equal to f(3)?

What is f(3)?

5. Jan 23, 2012

CJ256

Well for the first one if I plug in 2 where I have x all the equations are true except the x^3-3 so does that mean that it is not continuous even though it in the second equation 2=2?

For the second one I still have no clue where to start.

6. Jan 23, 2012

CJ256

Ok so for the first problem, I got that the function exists because of the piece wise function two of the functions equal 5 but I still need help with the second question

7. Jan 23, 2012

SammyS

Staff Emeritus
It's not asking if the function exists. It's asking if the limit exists.

It's quite clear from Post #5, that you don't understand the piecewise definition of a function. What are each of the following for the first function?
f(-1) =   ?

f(0) =   ?

f(1) =   ?

f(1) =   ?

f(1.9) =   ?

f(2) =   ?

f(2.1) =   ?

f(3) =   ?

f(4) =   ?