Results 1 to 5 of 5

Math Help - Continuous functions/Limits and continuity

  1. #1
    Newbie
    Joined
    Aug 2007
    Posts
    3

    Continuous functions/Limits and continuity

    Uhm, yeah, so I was studying for my test and realized that there's a lot I really didn't understand. This is one of those things:

    Instructions:

    Find a value for a so that the function is continuous.

    1. f(x) = {2x + 3, x 2; ax + 1, x > 2}

    2. f(x) = {4 - x^2, x < -1; ax^2 - 1, x >= 1}

    3. f(x) = {x^2 + x + a, x < 1; x^3, x >= 1}

    (This thing here: >= is a greater than or equals to sign since I couldn't figure out how to make it and I couldn't find anything to copy-paste off of the internet.)

    The other problem I had was a little different.

    Instructions:

    a) Draw the graph of f.
    b) At what points c in the domain of f does the limit exist?
    c) At what points c does only the left-hand limit exist?
    d) At what points c does only the right-hand limit exist?

    Problem:

    4. f(x) = {x, -1 ≤ x < 0, or 0 < x ≤ 1; 1, x=0; 0, x<-1, or x >1}

    My problem here is drawing the graph. It's the 'or's' that throw me off. I'm fairly certain that if I were to figure out the graph, that I could easily do the rest of the problem. I may be wrong, of course...

    (Also, if it's not too much trouble, and I'm sorry if it sounds a bit pretentious especially since I'm asking for help, but if you could maybe do part of it or explain how it goes and allow me to figure it out for myself? It is for a test, as I said, and I want to make sure that I perfectly understand it. But y'know, only if it's not too much trouble.)


    Also, I was wondering another thing, though this is just a general question or comment. Does it break the rules that I'm putting so many problems/questions in one post or is that still okay? I'm unsure.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,649
    Thanks
    1597
    Awards
    1
    Here is some help on #3. The graph is for a=0.
    The left-limit is 2+a at x=1; the right-limit is 1.
    Now what should we make a in order to make them equal?
    Attached Thumbnails Attached Thumbnails Continuous functions/Limits and continuity-temp.gif  
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Aug 2007
    Posts
    3
    Uhm, okay...so it looks like that graph would be continuous if the parabola were to move a little to the right. And if a=0 there, then should a=-1?

    It seems to me that I could get the answer by trial and error with my calculator, is that right? If so, do you know if there's a way I could determine the answer algebraically? I'm not sure that we're allowed to use a calculator.

    Though I remember, when hearing my classmates doing a similar problem, something about solving the top portion for x and then plugging x in? Or am I way off there?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,649
    Thanks
    1597
    Awards
    1
    Quote Originally Posted by Jane Doe View Post
    that graph would be continuous if the parabola were to move a little to the DOWN. And if a=0 there, then should a=-1?
    CORRECT!
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor red_dog's Avatar
    Joined
    Jun 2007
    From
    Medgidia, Romania
    Posts
    1,252
    Thanks
    5
    Quote Originally Posted by Jane Doe View Post
    Problem:

    4. f(x) = {x, -1 ≤ x < 0, or 0 < x ≤ 1; 1, x=0; 0, x<-1, or x >1}

    My problem here is drawing the graph. It's the 'or's' that throw me off. I'm fairly certain that if I were to figure out the graph, that I could easily do the rest of the problem. I may be wrong, of course...
    You can write the function in a more explicit way.
    f(x)=\left\{\begin{array}{cc}<br />
0, & x<-1\\<br />
x, & -1\leq x<0\\<br />
1, & x=0\\<br />
x, & 0<x\leq 1\\<br />
0, & x>1<br />
\end{array}\right.

    Now, can you draw the graph?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Continuity of inverse of a continuous function
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: May 3rd 2011, 11:28 PM
  2. Replies: 8
    Last Post: March 27th 2009, 04:23 AM
  3. Continuity - continuous functions
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 19th 2009, 03:05 AM
  4. Help in Limits of Continuity...
    Posted in the Calculus Forum
    Replies: 3
    Last Post: June 17th 2008, 09:09 AM
  5. Replies: 2
    Last Post: December 1st 2007, 02:30 PM

Search Tags


/mathhelpforum @mathhelpforum