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Math Help - Finding limits

  1. #1
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    Finding limits

    Could someone please help me find these limits? Been working for hours already
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  2. #2
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    Can you show your hours of working?

    Spoiler:
    \lim_{x \to 1 } \frac{x^2-1}{x-1}= \lim_{x \to 1 } \frac{(x-1)(x+1)}{x-1}=  \lim_{x \to 1 } x+1= 1+1 = 2
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  3. #3
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by letzdiscuss View Post
    Could someone please help me find these limits? Been working for hours already
    I'm numbering you're problems:

    1. 2.
    3. 4.
    5. 6.
    7.

    1.) Note that \frac{x}{|x|}=sign(x)=\left\{\begin{array}{lr}1&:x  >0\\0&:x=0\\-1&:x<0\end{array}\right\}

    Spoiler:
    So \frac{x^2}{|x|}=sign(x)\cdot x and \lim_{x\to0}(sign(x)\cdot x)=0


    --------

    2.) This function is continuous (from the right) at x=1.

    Spoiler:
    Just plug it in.


    --------

    3.) Just plug it in.

    --------

    4.) Start by factoring the numerator and see where that gets you.
    Spoiler:
    x^3-8=(x-2)(x^2+2x+4). Cancel out the (x-2) terms, and plug in x=2.


    --------

    5.) Same strategy as the last one.

    --------

    6.) As x gets small, \frac{1}{|x|} gets really big, so \lim_{x\to0}\frac{1}{|x|}=\infty.

    --------

    7.) Same method as (4) and (5).
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  4. #4
    Junior Member
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    Quote Originally Posted by redsoxfan325 View Post
    I'm numbering you're problems:



    1. 2.

    3. 4.

    5. 6.

    7.



    1.) Note that \frac{x}{|x|}=sign(x)=\left\{\begin{array}{lr}1&:x  >0\\0&:x=0\\-1&:x<0\end{array}\right\}



    Spoiler:
    So \frac{x^2}{|x|}=sign(x)\cdot x and \lim_{x\to0}(sign(x)\cdot x)=0




    --------



    2.) This function is continuous (from the right) at x=1.



    Spoiler:
    Just plug it in.




    --------



    3.) Just plug it in.



    --------



    4.) Start by factoring the numerator and see where that gets you.
    Spoiler:
    x^3-8=(x-2)(x^2+2x+4). Cancel out the (x-2) terms, and plug in x=2.




    --------



    5.) Same strategy as the last one.



    --------



    6.) As x gets small, \frac{1}{|x|} gets really big, so \lim_{x\to0}\frac{1}{|x|}=\infty.



    --------



    7.) Same method as (4) and (5).

    Thanks !! Got it !!
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