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Math Help - Asymptotes...help please

  1. #1
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    Asymptotes...help please

    I am unsure of where to start on these problems...I remember learning asymptotes, but I am a little confused by these problems....they seem different than what we learned...if someone could please help explain them to me, I would greatly appreciate it.

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    MHF Contributor red_dog's Avatar
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    16) \lim_{x\to\pm\infty}\frac{3}{x-9}=0\Rightarrow y=0 horizontal asymptote.

    17) f(x)=\frac{5x+7}{x^2+3x-10}=\frac{5x+7}{(x-2)(x+5)}
    \lim_{x\nearrow 2}f(x)=-\infty, \ \lim_{x\searrow 2}f(x)=\infty\Rightarrow x=2 vertical asymptote.

    Find the limits in x=-5 and you'll get another vertical asymptote.
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    Thank you very much for your help! but I don't quite remember doing this...I am still a little bit confused about what you did...
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    Quote Originally Posted by Raleigh View Post
    Thank you very much for your help! but I don't quite remember doing this...I am still a little bit confused about what you did...
    Hello,

    a few general remarks:

    an asymptote is a straight line or a curve which the graph of the function will approach when the |x|-values increases unlimited.
    With your examples you only have to calculate the limit of the values of the function if x approaches infinity.

    I assume that you know that \lim_{x \mapsto \infty}\left(\frac1x \right) = 0 and therefore

    \lim_{x \mapsto \infty}\left(\frac ax \right) = \lim_{x \mapsto \infty}\left(a \cdot \frac1x \right) = a \cdot \lim_{x \mapsto \infty}\left(\frac1x \right) = 0 and \lim_{x \mapsto \infty}\left(\frac1{x^n} \right) = 0~,~n \in \mathbb{N}

    Using this property the calculation of limits becomes easy:

    \lim_{x \mapsto \infty}\left(\frac{5x+7}{x^2+3x-10}  \right)=\lim_{x \mapsto \infty}\left(  \dfrac{x^2\left(\frac5x+\frac7{x^2}\right)} {x^2\left(1+\frac3x-\frac{10}{x^2}\right)}  \right) = \frac01 = 0

    That means the asymptote is here: y = 0
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