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Math Help - limit questions

  1. #1
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    limit questions

    I have two questions. The first one, I got 1 so I just want a confirmation that that is right (or then help me with what I did wrong)
    And the second I have no clue

    I need to find the limits if they exist (or show that they dont exist)
    a) limit as n approaches infinity of [1+(1/(2n))] ^ (n-1)

    b) lim as n approaches infinity of [ ln ( n + cos n)] / [ln n^2]

    for a) I figured 1/infinity is 0 so 1+0 is 1 and 1 to the power of anything is 1.

    WIll someone help me with b)
    Thank you!!
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  2. #2
    Senior Member DeMath's Avatar
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    Quote Originally Posted by portstar View Post
    I have two questions. The first one, I got 1 so I just want a confirmation that that is right (or then help me with what I did wrong)
    And the second I have no clue

    I need to find the limits if they exist (or show that they dont exist)
    a) limit as n approaches infinity of [1+(1/(2n))] ^ (n-1)

    Thank you!!
    I hope this decision for you is clear.

    \mathop {\lim }\limits_{n \to \infty } {\left( {1 + \frac{1}{{2n}}} \right)^{n - 1}} = \left\{ \begin{gathered}2n = k \Leftrightarrow n = \frac{k}{2} \hfill \\n \to \infty ,{\text{ }}k \to \infty  \hfill \\ \end{gathered}  \right\}=

    = \mathop {\lim }\limits_{k \to \infty } {\left( {1 + \frac{1}{k}} \right)^{\frac{k}{2} - 1}} = \mathop {\lim }\limits_{k \to \infty } \left\{ {\sqrt {{{\left( {1 + \frac{1}{k}} \right)}^k}} :\left( {1 + \frac{1}{k}} \right)} \right\} =

    = \mathop {\lim }\limits_{k \to \infty } \sqrt {{{\left( {1 + \frac{1}{k}} \right)}^k}} :\mathop {\lim }\limits_{k \to \infty } \left( {1 + \frac{1}{k}} \right) = \sqrt e :1 = \sqrt e .
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  3. #3
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    Quote Originally Posted by portstar View Post
    I need to find the limits if they exist (or show that they dont exist)
    a) limit as n approaches infinity of [1+(1/(2n))] ^ (n-1)
    for a) I figured 1/infinity is 0 so 1+0 is 1 and 1 to the power of anything is 1.
    You need a lot more work on #1.
    This is true in general. \left( {1 + \frac{a}{{n + b}}} \right)^{cn}  \to e^{ac} .

    Now write \left( {1 + \frac{1}<br />
{{2n}}} \right)^{n - 1}  = \frac{{\left( {1 + \frac{{1/2}}<br />
{n}} \right)^n }}<br />
{{\left( {1 + \frac{1}<br />
{{2n}}} \right)}} \to ?
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  4. #4
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    darn, I wished it would have been that easy ! haha thanks though
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  5. #5
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    Quote Originally Posted by portstar View Post
    I have two questions. The first one, I got 1 so I just want a confirmation that that is right (or then help me with what I did wrong)
    And the second I have no clue

    I need to find the limits if they exist (or show that they dont exist)
    a) limit as n approaches infinity of [1+(1/(2n))] ^ (n-1)

    b) lim as n approaches infinity of [ ln ( n + cos n)] / [ln n^2]

    for a) I figured 1/infinity is 0 so 1+0 is 1 and 1 to the power of anything is 1.

    WIll someone help me with b)
    Thank you!!
    It appears that these questions might come from an assignment that contributes towards your final grade. Thread closed.
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