Results 1 to 10 of 10

Math Help - infinite/finite limits

  1. #1
    Member
    Joined
    Feb 2009
    Posts
    106

    infinite/finite limits

    which of these limits is finite?

    A) lim 1.001^n
    n->infinity

    B) lim ln (1/1+t)
    t->infinity

    C) lim (e^x)/x
    x->infinity

    D) lim (1 + x/2)^(1/x)
    x->0

    i'm sorry that they're not in the nicest format
    i figured that the first two were just infinity, but i'm not sure..
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,513
    Thanks
    1404
    Quote Originally Posted by buttonbear View Post
    which of these limits is finite?

    A) lim 1.001^n
    n->infinity

    B) lim ln (1/1+t)
    t->infinity

    C) lim (e^x)/x
    x->infinity

    D) lim (1 + x/2)^(1/x)
    x->0

    i'm sorry that they're not in the nicest format
    i figured that the first two were just infinity, but i'm not sure..
    The first three are infinite.

    A) Any number > 1 \to \infty as it gets exponentiated.

    B) The inside of the brackets \to \frac{1}{\infty} \to 0.

    \log{x} \to -\infty as x \to 0.

    C) The top and bottom both tend to \infty. This is indeterminate, but we can use L'Hospital.

    \lim_{x \to \infty}\frac{e^x}{x} = \lim_{x \to \infty}\frac{\frac{d}{dx}e^x}{\frac{d}{dx}x} = \lim_{x \to \infty}\frac{e^x}{1} \to \infty.

    Not sure about D.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,361
    Thanks
    39
    If the last one is

    \lim_{x \to 0} \left( 1 + \frac{x}{2} \right)^{\frac{1}{x}}

    then

     \lim_{x \to 0} e^{ \ln \left( 1 + \frac{x}{2} \right)^{\frac{1}{x}}} = \lim_{x \to 0} e^{ \frac{1}{x}\ln \left( 1 + \frac{x}{2} \right)}

    which comes down to finding

     \lim_{x \to 0} \frac{1}{x}\ln \left( 1 + \frac{x}{2} \right) = \frac{1}{2}.

    using L'Hopital's rule, so

    \lim_{x \to 0} \left( 1 + \frac{x}{2} \right)^{\frac{1}{x}} = <br />
\sqrt{e}
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Feb 2009
    Posts
    106
    i'm a little confused by what happens in the last two lines of the last limit- would you mind explaining?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,361
    Thanks
    39
    Quote Originally Posted by buttonbear View Post
    i'm a little confused by what happens in the last two lines of the last limit- would you mind explaining?
    Sure. Using L'Hoptials rule

    \lim_{x \to 0} \frac{1}{x}\ln \left( 1 + \frac{x}{2} \right) = \lim_{x \to 0} \frac{1}{1 + \frac{x}{2} } \frac{1}{2} = \frac{1}{2}

    so

    \lim_{x \to 0} e^{ \ln \left( 1 + \frac{x}{2} \right)^{\frac{1}{x}}} = \lim_{x \to 0} e^{ \frac{1}{x}\ln \left( 1 + \frac{x}{2} \right) }= e^{ \lim_{x \to 0} \frac{1}{x}\ln \left( 1 + \frac{x}{2} \right) }= e^{1/2}
    Last edited by Jester; March 10th 2009 at 06:45 AM.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member
    Joined
    Feb 2009
    Posts
    106
    \lim_{x \to 0} \frac{1}{x}\ln \left( 1 + \frac{x}{2} \right) = \frac{1}{2}

    so you used l'hopital's rule here?
    i understand everything else..i'm just not getting the 1/2
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    11,623
    Thanks
    428
    Quote Originally Posted by buttonbear View Post
    \lim_{x \to 0} \frac{1}{x}\ln \left( 1 + \frac{x}{2} \right) = \frac{1}{2}

    so you used l'hopital's rule here?
    i understand everything else..i'm just not getting the 1/2

    \lim_{x \to 0} \frac{\ln \left( 1 + \frac{x}{2} \right)}{x}

    L'Hopital ... \lim_{x \to 0} \frac{f'(x)}{g'(x)} ...

    \lim_{x \to 0} \frac{\frac{\frac{1}{2}}{1 + \frac{x}{2}}}{1} = \frac{1}{2}
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Member
    Joined
    Feb 2009
    Posts
    106
    i guess sometimes i just need things spelled out for me. thanks so much!
    Follow Math Help Forum on Facebook and Google+

  9. #9
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,361
    Thanks
    39
    Quote Originally Posted by buttonbear View Post
    \lim_{x \to 0} \frac{1}{x}\ln \left( 1 + \frac{x}{2} \right) = \frac{1}{2}

    so you used l'hopital's rule here?
    i understand everything else..i'm just not getting the 1/2
    If f = \ln \left(1 + \frac{x}{2} \right)\, \text{and}\; g = x then f' = \frac{1}{1 + \frac{x}{2}} \cdot \frac{1}{2}\; \text{and}\; g' = 1 (the first by the chain rule). so

    \lim_{x \to 0} \frac{f'}{g'} = \lim_{x \to 0} \frac{1}{1 + \frac{x}{2}} \cdot \frac{1}{2} = \frac{1}{2}

    This is where the 1/2 comes from. Better?
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Member
    Joined
    Feb 2009
    Posts
    106
    much better- thanks so much!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Finite and infinite sets
    Posted in the Discrete Math Forum
    Replies: 11
    Last Post: August 6th 2011, 03:53 PM
  2. Finite, countably infinite, or uncountable
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: March 25th 2011, 08:40 AM
  3. finite or infinite?
    Posted in the Discrete Math Forum
    Replies: 5
    Last Post: September 3rd 2010, 09:39 PM
  4. Finite and Infinite order
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 6th 2010, 07:51 AM
  5. Finite>Infinite
    Posted in the Advanced Math Topics Forum
    Replies: 9
    Last Post: February 20th 2006, 11:37 AM

Search Tags


/mathhelpforum @mathhelpforum