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Math Help - [SOLVED] Natural Log Limit

  1. #1
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    [SOLVED] Natural Log Limit

    Evaluate
    \lim_{t\to-5} \ln(t+5)^2

    Since  \ln(0)^2 is undefined, I used L'Hopital's Rule twice to get

     \lim_{t\to-5} \ln(t+5)^2 = \lim_{t\to-5}\frac{2 \ln(t+5)}{t+5} = \frac{0}{0}

    This gives an indeterminate case as well. So I used L'Hopital's Rule once again to get

    \lim_{t\to-5}\frac{2 \ln(x+5)-1}{(x+5)^2} = \frac{0}{0}

    which brings be back to an indeterminate case because of the x+5 in the denominator irregardless of how many times I take the derivative.

    I know that  \lim_{x\to0+} \ln(x) \to -\infty , but don't know how to apply it here.

    Graphing it on my calculator, I see that the limit goes to positive infinity.
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  2. #2
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    Quote Originally Posted by Paperwings View Post
    Evaluate
    \lim_{t\to-5} \ln(t+5)^2

    Since  \ln(0)^2 is undefined, I used L'Hopital's Rule twice to get

     \lim_{t\to-5} \ln(t+5)^2 = \lim_{t\to-5}\frac{2 \ln(t+5)}{t+5} = \frac{0}{0}

    This gives an indeterminate case as well. So I used L'Hopital's Rule once again to get

    \lim_{t\to-5}\frac{2 \ln(x+5)-1}{(x+5)^2} = \frac{0}{0}

    which brings be back to an indeterminate case because of the x+5 in the denominator irregardless of how many times I take the derivative.

    I know that  \lim_{x\to0+} \ln(x) \to -\infty , but don't know how to apply it here.

    Graphing it on my calculator, I see that the limit goes to positive infinity.
    I am not 100% sure about this, but I think that ln(0)^2 does not count as an indeterminate form. There are a specific set of situations and they all involve more than one term. See this Wiki page on it:

    Indeterminate form - Wikipedia, the free encyclopedia

    Now as for your limit, remember that ln(x) is not defined for x \le 0. You have a horizontal shift from ln(x) but the concept still applies. When you graph your equation you should not see anything from x=-5 and below, thus the limit from the left hand side DNE. That is not a rigorous proof of course but it sounds like you more so need an explanation than proof.
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  3. #3
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    Thank you. The limit doesn't exist because the one-sided limit from the left doesn't exist.
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