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Math Help - Backwards limit problem.

  1. #1
    Junior Member beebe's Avatar
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    Backwards limit problem.

    If \lim_{x\to1}\frac{f(x)-8}{x-1}=10 then find \lim_{x\to1}f(x)

    This has me pretty stumped. Since the denominator is zero at x=1, I can't use limit of quotient is the quotient of the limits. My only other thought was to figure out f(x), which should be a polynomial where f(x)-8 has a factor of 1-x, but I can't figure out what that would be.
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  2. #2
    A Plied Mathematician
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    Re: Backwards limit problem.

    Since the original limit is finite, what must

    \lim_{x\to 1}(f(x)-8) be?
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  3. #3
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    Re: Backwards limit problem.

    To do this problem, we're going to find out what f(x) is, and then take the limit. To find f(x), we do this:

    We know that f(x)-8 has to have an (x-1) in it. We also know that the other factor must be equal to 10, when we plug in a 1 for x. Therefore, the other factor must be (x+9)

    If we FOIL, we have:

    (x-1)(x+9) = x^2+8x-9

    Now we separate the -8 from the mix:

    x^2+8x-1-8

    So we have:

    f(x) = x^2+8x-1

    When we plug in 1, we get our answer: 8

    I hope this helps!

    -Nathan
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  4. #4
    A Plied Mathematician
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    Re: Backwards limit problem.

    Quote Originally Posted by TutorMeNate View Post
    To do this problem, we're going to find out what f(x) is, and then take the limit.
    I'm not sure I agree with this approach. f(x) doesn't even have to be defined in order to solve this problem.
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  5. #5
    Junior Member beebe's Avatar
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    Re: Backwards limit problem.

    Quote Originally Posted by Ackbeet View Post
    Since the original limit is finite, what must

    \lim_{x\to 1}(f(x)-8) be?
    I'm guessing 8... So if we evaluated \frac{f(x)-8}{x-1} by just plugging in x=1, should we get a fraction of \frac{0}{0} because there is a finite limit?
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  6. #6
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    Re: Backwards limit problem.

    I'm not sure I agree with this approach. f(x) doesn't even have to be defined in order to solve this problem.
    I figured there was another way to solve this problem, but to be honest, I did not know the exact method, so I used a way that I knew would work.

    Sometimes the front door is locked, so you need to break through the window

    -Nathan
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  7. #7
    A Plied Mathematician
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    Re: Backwards limit problem.

    Quote Originally Posted by beebe View Post
    I'm guessing 8... So if we evaluated \frac{f(x)-8}{x-1} by just plugging in x=1, should we get a fraction of \frac{0}{0} because there is a finite limit?
    Looks good to me!
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  8. #8
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    Re: Backwards limit problem.

    Quote Originally Posted by TutorMeNate View Post
    To do this problem, we're going to find out what f(x) is, and then take the limit. To find f(x), we do this:

    We know that f(x)-8 has to have an (x-1) in it.
    No, we don't know that because we don't know that f is a polynomial.

    We also know that the other factor must be equal to 10, when we plug in a 1 for x. Therefore, the other factor must be (x+9)

    If we FOIL, we have:

    (x-1)(x+9) = x^2+8x-9

    Now we separate the -8 from the mix:

    x^2+8x-1-8

    So we have:

    f(x) = x^2+8x-1

    When we plug in 1, we get our answer: 8

    I hope this helps!

    -Nathan
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  9. #9
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    Re: Backwards limit problem.

    Good except for the "I'm guessing" part! No need to guess!
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  10. #10
    Junior Member beebe's Avatar
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    Re: Backwards limit problem.

    Thanks!
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  11. #11
    Grand Panjandrum
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    Re: Backwards limit problem.

    Quote Originally Posted by TutorMeNate View Post
    To do this problem, we're going to find out what f(x) is, and then take the limit. To find f(x), we do this:

    We know that f(x)-8 has to have an (x-1) in it. We also know that the other factor must be equal to 10, when we plug in a 1 for x. Therefore, the other factor must be (x+9)

    If we FOIL, we have:

    (x-1)(x+9) = x^2+8x-9

    Now we separate the -8 from the mix:

    x^2+8x-1-8

    So we have:

    f(x) = x^2+8x-1

    When we plug in 1, we get our answer: 8

    I hope this helps!

    -Nathan
    Wrong! Where does it say f(x) is a polynomial. We do not need to assume a polynomial to do this

    Sorry did not see Hall's posts. Note to self: No more shooting from the hip (it don't harf hurt)

    CB
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  12. #12
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    Re: Backwards limit problem.

    Oops, I apologize, I read the problem and started thinking polynomial:

    My only other thought was to figure out f(x), which should be a polynomial where f(x)-8 has a factor of 1-x, but I can't figure out what that would be.
    I should have thought it through more. Thanks for the correction!

    -Nathan
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