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Math Help - answer of ∞ - ln∞

  1. #1
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    answer of ∞ - ln∞

    hi do anyone knows what ∞ - ln∞ gives?




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  2. #2
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    It gives infinity. ln∞ does tend to infinity, but not very fast.

    Keep in mind though, that infinity isn't like any ordinary number. ∞+∞ = ∞, for example.
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  3. #3
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    Quote Originally Posted by firebrend View Post
    hi do anyone knows what ∞ - ln∞ gives?




    [IMG]file:///C:/Users/Arwin/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif[/IMG][IMG]file:///C:/Users/Arwin/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif[/IMG]
    Please post the actual question, exactly as it's written.
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  4. #4
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    Quote Originally Posted by firebrend View Post
    hi do anyone knows what ∞ - ln∞ gives?
    \infty- ln\infty doesn't give anything. "Infinity" is not a real number and ordinary functions such as "logarithm" and "subtraction" do not apply to it. There are extended number systems in which "infinity" is defined as a number but they still do not apply "logarithm" or "subtraction" in the usual sense.

    Something of the form lim [f(x)- ln(g(x))] where lim f= \infty and lim g= \infty may exist but the limit will depend upon exactly how f and g "go to infinity".

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  5. #5
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    The question is:
    given: f(x)= x-3/2 ln⁡〖(x^2+2) 〗

    Using mathematical arguments, investigate what happens to f(x) as x -> ∞
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  6. #6
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    \displaystyle f(x) = x - \frac{3}{2}\ln{(x^2+2)}


    \displaystyle f'(x) = 1 - \frac{3x}{x^2 + 2} \to 1 as x \to \infty.

    Since the derivative tends to a positive number, the function will continue to increase without bound.


    So \displaystyle \lim_{x \to \infty} f(x) = \infty.
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  7. #7
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    Hello, firebrend!

    Here is a really sloppy solution . . .


    \text{Given: }\:f(x) \:=\:x - \tfrac{3}{2}\ln(x^2+2)

    \displaystyle \text{Investigate: }\;\lim_{x\to\infty}f(x)

    We have: . f(x) \:=\:x - \ln(x^2+2)^{\frac{3}{2}}


    For very large \,x,\:f(x) \:\approx\:x - \ln(x^2)^{\frac{3}{2}} \;=\;x - \ln(x^3)

    Then: . f(x) \;\approx\;\ln(e^x) - \ln(x^3) \;=\;\ln\left(\dfrac{e^x}{x^3}\right)


    It can be shown that: . \displaystyle \lim_{x\to\infty}\,\ln\left(\dfrac{e^x}{x^3}\right  ) \;=\;\infty

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