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Math Help - Limits for a Piecewise function

  1. #1
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    Limits for a Piecewise function

    Of the piecewise below, (a) find the right and left hand limit of f at x= -1. (b) Is f continuous at x= -1? Explain.

    No program, meaning using a program graphing Someone else's computer, who I may not download things onto. Though, the function is straightforward.

    1, x < or = to -1
    -x, -1 < x < 0
    1, x= 0
    -x, 0 < x < 1
    1, x > or = to 1


    Thought:
    (a) right: -1
    left: -1

    (b) I think so. Even though there are two points of discontinuity, they are removable points.

    For extra help, how does one get rid of removable points of discontinuity?

    Thanks!
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  2. #2
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    Quote Originally Posted by Truthbetold View Post
    Of the piecewise below, (a) find the right and left hand limit of f at x= -1. (b) Is f continuous at x= -1? Explain.

    No program, meaning using a program graphing Someone else's computer, who I may not download things onto. Though, the function is straightforward.

    1, x < or = to -1
    -x, -1 < x < 0
    1, x= 0
    -x, 0 < x < 1
    1, x > or = to 1


    Thought:
    (a) right: -1
    left: -1

    (b) I think so. Even though there are two points of discontinuity, they are removable points.

    For extra help, how does one get rid of removable points of discontinuity?

    Thanks!
    Hello,

    The limits are \lim_{x\to -\infty}f(x) = 1 and \lim_{x\to +\infty}f(x) = 1

    To prove if f is continuous at x = 1 use the definition:

    A function f is continuous at x = a iff

    \lim_{x\to a \wedge x<a}f(x)=\lim_{x\to a \wedge x>a}f(x)=f(a)

    With your problem:
    f(x)=\left\{\begin{array}{lr}1&x\leq-1\\-x&-1<x<0\\1&x=0 \\-x&0<x<1\\1&x\geq1\end{array}\right.

    You know a = -1

    \underbrace{\lim_{x\to (-1) \wedge x<(-1)}f(x)}_{\text{use first row}}=\underbrace{\lim_{x\to (-1) \wedge x>(-1)}f(x)}_{\text{use second row}}=\underbrace{f(-1)}_{\text{first row}}=1 Therefore the function is continuous at x = -1
    Attached Thumbnails Attached Thumbnails Limits for a Piecewise function-stckw_fkt.gif  
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