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Math Help - Limits to Infinity and Slant Asymptotes

  1. #1
    Junior Member Khaali91's Avatar
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    Exclamation Limits to Infinity and Slant Asymptotes

    1. find lim(as x approaches positive infinity) of
    2x+xsin(3x)
    5x^2-2x+1




    2. Find the horizontal,vertical, and slant asymptotes of:
    y= x^3-X^2+10
    3x^2-4X





    Any help would be greatly appreciated, Thank YOU
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  2. #2
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    Hello, Khaali91!

    1. Find the horizontal, vertical, and slant asymptotes of:

    y \;=\;\frac{x^3-x^2+10}{3x^2-4x}

    For a horizontal asymptote, consider \lim_{x\to\infty} y

    Divide top and bottom by x^2\!:\quad \frac{\dfrac{x^3}{x^2} - \dfrac{x^2}{x^2} + \dfrac{10}{x^2}}{\dfrac{3x^2}{x^2} - \dfrac{4x}{x^2}} \;=\;\frac{x - 1 + \dfrac{10}{x^2}}{3-\dfrac{4}{x}}

    Then: . \lim_{x\to\infty}\frac{x - 1 + \frac{10}{x^2}}{3-\frac{4}{x}} \;=\; \frac{\infty - 1 + 0}{3 - 0} \;=\;\infty

    . . There is no horizontal asymptote.



    For vertical asymptotes, determine when the denominator equals zero.

    We have: . x(3x+4) \:=\:0\quad\Rightarrow\quad x \;=\;0,\:\frac{4}{3}

    The vertical asymptotes are: . x\:=\:0\text{ (y-axis)},\;x \:=\: \frac{4}{3}



    For slant asymptotes, consider \lim_{x\to\infty} y

    We have: . \lim_{x\to\infty}\frac{x^3 - x^2 + 10}{3x^2-4x} \;=\;\lim_{x\to\infty}\frac{x - 1 + \frac{10}{x^2}}{3 - \frac{4}{x}} \;=\;\frac{x - 1 + 0}{3 - 0} \;=\;\frac{x-1}{3}


    Therefore, the slant asymptote is: . y \;=\;\frac{1}{3}x - \frac{1}{3}

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  3. #3
    Junior Member Khaali91's Avatar
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    Thank you!

    For the 1st one where you said the assume the limit is to infinity for the horizontal asymptote, can that be applied to all horizontal asymptotes?

    THANK YOU
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  4. #4
    Junior Member Khaali91's Avatar
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    Exclamation help plz?

    can anyone still help me with

    1. find the limit (as x approaches positive infinity) of
    2x+xsin(3x)
    5x^2-2x+1
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  5. #5
    Math Engineering Student
    Krizalid's Avatar
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    Divide top & bottom by x^2.
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  6. #6
    Junior Member Khaali91's Avatar
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    Thank You

    but would the limit of sin3x/x still equal out to 0? thats where i got confused =/
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  7. #7
    Math Engineering Student
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    Yes.
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  8. #8
    Junior Member Khaali91's Avatar
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    Talking YAY

    THANK YOUUUUU! lol

    YAY im good to go
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