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Thread: How do I find the horizontal asymptote of a exponential function?

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    How do I find the horizontal asymptote of a exponential function?

    I am given a problem and I am asked to find the asymptotes after I have graphed it.

    Here is the function:

    f(x) = (e^x) +2


    I am having trouble trying to figure out how it would become undefined along the y-axis. I thought that perhaps I should set the y equal to zero and subtract the 2 out and then take the natural log of both sides to drop the x out of the exponent area. That would give me a negative number to work out with the natural log. Unfortunately, I don't think this is the right way to go about it. I looked in the back of the sheet and the answer for a horizontal asymptote was given as y=2. I believe I may be making this more complicated than it needs to be.

    Thanks for looking this over and I appreciate the help.
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    Re: How do I find the horizontal asymptote of a exponential function?

    horizontal asymptotes describe the end behavior of the function $f(x)$ in one or both directions ...

    as $x \to +\infty$, $f(x) = e^x+2$ increases w/out bound

    what about in the other direction?

    $\displaystyle \lim_{x \to -\infty} e^x + 2 = ?$
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    Re: How do I find the horizontal asymptote of a exponential function?

    Looking at the function, I now see that the parent is e^x. So (e^x)+2 looks like a basic vertical shift of +2. Normally the basic exponential function would approach zero on the y axis with a smaller and smaller fraction as it's output. I am not quite into limits yet, but is this correct?
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    Re: How do I find the horizontal asymptote of a exponential function?

    as $x \to -\infty$, $f(x)=e^x+2 \to 2 \implies$ the horizontal asymptote is $y=2$
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    Re: How do I find the horizontal asymptote of a exponential function?

    Quote Originally Posted by skeeter View Post
    as $x \to -\infty$, $f(x)=e^x+2 \to 2 \implies$ the horizontal asymptote is $y=2$
    He just stated he hasn't done limits yet.

    Without limits, one way would be to find values where the function is undefined. Start by taking the natural log of both sides.
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    Re: How do I find the horizontal asymptote of a exponential function?

    Quote Originally Posted by Awesome31312 View Post
    He just stated he hasn't done limits yet.

    Without limits, one way would be to find values where the function is undefined. Start by taking the natural log of both sides.
    The given exponential function is defined for all reals. I believe you mean find the vertical asymptote for the inverse function.

    Ok, Mr. Awesome, show him how it's done ...
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    Re: How do I find the horizontal asymptote of a exponential function?

    Quote Originally Posted by skeeter View Post
    The given exponential function is defined for all reals. I believe you mean find the vertical asymptote for the inverse function.

    Ok, Mr. Awesome, show him how it's done ...

    Hmm, you are right. I apologize for the miscommunication.

    Before learning limits, the elementary method is to use transformations, so:

    Observing the graph of y = e ^ (x) , we notice a horizontal asymptote: y = 0

    When the graph " y = e ^ (x) + 2 " is observed, the horizontal asymptote has been translated 2 units upwards on the y-axis. 0 + 2 = 2. Therefore, the horizontal asymptote is y = 0. The proof of this would be through limits. I strongly recommend you learn limits.
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    Re: How do I find the horizontal asymptote of a exponential function?

    Quote Originally Posted by Awesome31312 View Post
    When the graph " y = e ^ (x) + 2 " is observed, the horizontal asymptote has been translated 2 units upwards on the y-axis. 0 + 2 = 2. Therefore, the horizontal asymptote is y = 0.
    Therefore, the horizontal asymptote for   \ \ y \ = \ e^x + 2 \ \  is  \ \   y = 2 .
    Last edited by greg1313; Apr 18th 2017 at 03:16 PM.
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