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Math Help - Need help with a simple continuity problem.

  1. #1
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    Need help with a simple continuity problem.

    f(x) = { cos(x-2)+1 x is less than or equal to 2
    { sin(x-2)+2 2<x

    Is it continuous at x=2 at the right, left, both, or neither?

    I've tried to work it out and graph but i'm pretty confused right now. I think that it's not continuous because both pieces are not equal, but I dont know if it's right, left, or none.

    Thanks alot in advance!
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  2. #2
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    Re: Need help with a simple continuity problem.

    Quote Originally Posted by poipoipoi10 View Post
    f(x) = { cos(x-2)+1 x is less than or equal to 2
    { sin(x-2)+2 2<x

    Is it continuous at x=2 at the right, left, both, or neither?

    I've tried to work it out and graph but i'm pretty confused right now. I think that it's not continuous because both pieces are not equal, but I dont know if it's right, left, or none.
    \lim_{x \to 2^-} \cos(x-2)+1 = 2

    \lim_{x \to 2^+} \sin(x-2)+2 = 2

    f(2) = 2

    what's that tell you?
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  3. #3
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    Re: Need help with a simple continuity problem.

    Quote Originally Posted by poipoipoi10 View Post
    f(x) = { cos(x-2)+1 x is less than or equal to 2
    { sin(x-2)+2 2<x

    Is it continuous at x=2 at the right, left, both, or neither?
    Here is your function.
    f(x) = \left\{ {\begin{array}{rl}   {\cos (x - 2) + 1,} & {x \leqslant 2}  \\   {\sin (x - 2) + 2,} & {x > 2}  \\ \end{array} } \right.

    Here is some analysis notation: f(a + ) = \lim _{x \to a^ +  } f(x).
    There is of course a similar statement for f(a-).

    Now you need to see if f(2-)=f(2+).
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