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Math Help - Evaluating and simplifying limits

  1. #1
    Member realintegerz's Avatar
    Joined
    Aug 2008
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    Evaluating and simplifying limits

    I need help on this limit problem, we aren't given an equation....so I'm not really sure how to do these. We were supposed to have a lesson on limits but we didn't finish the lesson and our teacher let us wander off without knowing 1/2 the material

    Let f(x) =

    -x, x < - 1
    x -2, -1 less than or equal to x < 1
    1, x = 1
    (x-1)^2, x>1

    A) limit of f(x), as x approaches -1 from below
    B) limit of f(x), as x approaches -1 from above

    And also the same for 0 and 1

    I also need help on simplifying these limits, they are what they look like. The problems were so that the x would approach whatever number that would make the denominator 0. And I couldn't see any way to simplify these but if anyone here can that would be great.

    1) [(x/x-2) - x+2]/x-4

    2) [sq. root (1 + h) + 1]/h

    3) [(1/sq. root(4+h)) - 1/2]/h

    4) [sq. root (x-1)]/x^2 - 1

    5) [sq. root(x+1) - sq. root(2x-2)]/x-3
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  2. #2
    Super Member

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    Hello, realintegerz!

    The first problem is straight-forward.
    Exactly where is your difficulty?


    Let: . f(x) \:=\:\begin{Bmatrix} -x &  x < \text{-}1 \\<br />
x -2 & \text{-}1 \le x < 1 \\<br />
1 & x = 1 \\<br />
(x-1)^2 & x>1 \end{Bmatrix}


    a)\;\;\lim_{x\to\,\text{-}1^\text{-}}f(x)
    \lim_{x\to\,\text{-}1^-}f(x) \;=\;\lim_{x\to\,\text{-}1^-}(-x) \;=\;-(\text{-}1) \;=\;1



    b)\;\;\lim_{x\to\text{-}1^+}f(x)
    \lim_{x\to\,\text{-}1^+}f(x) \;=\;\lim_{x\to\,\text{-}1^+}(x-2) \;=\;\text{-}1 - 2 \:=\:-3



    c)\;\;\lim_{x\to0}f(x)
    \lim_{x\to0}f(x) \;=\;\lim_{x\to0}(x-2) \;=\;0-2 \;=\;-2



    \lim_{x\to1^-}f(x)
    \lim_{x\to1^-}f(x) \;=\;\lim_{x\to1^-}(x-2) \;=\;1-2 \;=\;-1



    e)\;\;\lim_{x\to1^+}f(x)
    \lim_{x\to1^+}f(x) \;=\;\lim_{x\to1^+}(x-1)^2 \;=\;(1-1)^2 \;=\;0


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


    A sketch makes the limits quite clear . . .
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