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Math Help - problems of limit at infinity

  1. #1
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    problems of limit at infinity

    Dear all,

    I have two similar questions but I am not getting the idea how to solve it the second one, is the approach in first part is correct or not???

    1.  \mathop {\lim }\limits_{x \to  + \infty } {\left( {x + \frac{2}{x}} \right)^{3x}}
    Let y = {\left( {x + \frac{2}{x}} \right)^{3x}}, then
    \mathop {\lim }\limits_{x \to  + \infty } y = \mathop {\lim }\limits_{x \to  + \infty } {e^{\ln {{\left( {x + \frac{2}{x}} \right)}^{3x}}}}

     = {e^{3x.\ln \left( {x + \frac{2}{x}} \right)}} = {e^{\mathop {\lim }\limits_{x \to  + \infty } 3x.\ln \left( {x + \frac{2}{x}} \right)}}

     = {e^{ + \infty . + \infty }} =  + \infty

    Am I right here???

    and Second one is
    2. \mathop {\lim }\limits_{x \to  - \infty } {\left( {\left| x \right| + \frac{2}{x}} \right)^{3x}}
    how to proceed here???

    Thanks in Advance.
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  2. #2
    MHF Contributor
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    That seems like an awful lot of work.

    On #1, the part in the parentheses is x + 2/x. This clearly increases without bound. Unless something is happening to control it, we have divergence. For example, the exponenent might be (1/(3x)) or (-x). If it clearly diverges, don't go to a whole lot of effort.

    See if this helps indirectly with #2.
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