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Math Help - Limit: Indeterminate Form/L'Hospital's Rule

  1. #1
    Member RedBarchetta's Avatar
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    Limit: Indeterminate Form/L'Hospital's Rule

    Evaluate the limit. (If you need to use - or , enter -INFINITY or INFINITY.)

    <br />
\mathop {\lim }\limits_{x \to \infty } \frac{{\ln \left( {\ln \left( {4x} \right)} \right)}}<br />
{{4x}}<br />

    Here's what I tried. Here's our first rule:

    <br />
\begin{gathered}<br />
  \mathop {\lim }\limits_{x \to \infty } \frac{{f(x)}}<br />
{{g(x)}} = \mathop {\lim }\limits_{x \to \infty } \frac{{f'(x)}}<br />
{{g'(x)}} \hfill \\<br />
  \mathop {\lim }\limits_{x \to \infty } \frac{{\tfrac{1}<br />
{{\ln 4x}} \cdot \tfrac{1}<br />
{{4x}} \cdot \tfrac{4}<br />
{1}}}<br />
{4} \hfill \\<br />
  \mathop {\lim }\limits_{x \to \infty } \frac{{\tfrac{1}<br />
{{x\ln 4x}}}}<br />
{4} \hfill \\ <br />
\end{gathered} <br />

    <br />
\begin{gathered}<br />
  \mathop {\lim }\limits_{x \to \infty } \frac{{\tfrac{1}<br />
{{x\ln 4x}}}}<br />
{4} \hfill \\<br />
  \mathop {\lim }\limits_{x \to \infty } \frac{1}<br />
{{4x\ln 4x}} \hfill \\ <br />
\end{gathered} <br />

    So then what? It's not infinity or -infinity.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
    Joined
    May 2008
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    Chicago, IL
    Posts
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    3
    Quote Originally Posted by RedBarchetta View Post
    Evaluate the limit. (If you need to use - or , enter -INFINITY or INFINITY.)

    <br />
\mathop {\lim }\limits_{x \to \infty } \frac{{\ln \left( {\ln \left( {4x} \right)} \right)}}<br />
{{4x}}<br />

    Here's what I tried. Here's our first rule:

    <br />
\begin{gathered}<br />
  \mathop {\lim }\limits_{x \to \infty } \frac{{f(x)}}<br />
{{g(x)}} = \mathop {\lim }\limits_{x \to \infty } \frac{{f'(x)}}<br />
{{g'(x)}} \hfill \\<br />
  \mathop {\lim }\limits_{x \to \infty } \frac{{\tfrac{1}<br />
{{\ln 4x}} \cdot \tfrac{1}<br />
{{4x}} \cdot \tfrac{4}<br />
{1}}}<br />
{4} \hfill \\<br />
  \mathop {\lim }\limits_{x \to \infty } \frac{{\tfrac{1}<br />
{{x\ln 4x}}}}<br />
{4} \hfill \\ <br />
\end{gathered} <br />

    <br />
\begin{gathered}<br />
  \mathop {\lim }\limits_{x \to \infty } \frac{{\tfrac{1}<br />
{{x\ln 4x}}}}<br />
{4} \hfill \\<br />
  \mathop {\lim }\limits_{x \to \infty } \frac{1}<br />
{{4x\ln 4x}} \hfill \\ <br />
\end{gathered} <br />

    So then what? It's not infinity or -infinity.
    \lim_{x\to{\infty}}\frac{\ln(\ln 4x)}{4x}

    \lim_{x\to{\infty}}\frac{\frac{1}{x\ln 4x}}{4}

    \lim_{x\to{\infty}}\frac{1}{4x\ln 4x}=\color{red}\boxed{0}. This the case because the denominator approaches infinity : \frac{1}{\infty}\rightarrow 0.
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