Need Help Determining Continuity of Functions

[CJ256]

Homework Statement

Homework Equations

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

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.

 

[SammyS]

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

If so, what is that definition?

 

[CJ256]

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

If so, what is that definition?

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:

 

[SammyS]

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:

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)?

 

[CJ256]

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)?

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.

 

[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

 

[SammyS]

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

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) =   ?  
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