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Help on limits

  1. Nov 26, 2008 #1
    Hi

    I´m new here and i´m new on this course. I have a test tomorrow and i need to know how to calculate limits, but i have some that i can´t solve, please can you solve it, ins´t homework, it´s only to learn (don´t use l'hospital):

    lim | x | / (x + 1)
    x->0


    lim (1 + sin x)^(1/x)
    x->0


    lim (e^(2 sin x) - e^(sin x)) / (sin 2x)
    x->0


    Thanks in advance
     
  2. jcsd
  3. Nov 26, 2008 #2

    lurflurf

    User Avatar
    Homework Helper

    lim | x | / (x + 1)
    x->0

    Show the function is continuous at 0
    limit of a continous function is an evaluation

    lim (1 + sin x)^(1/x)
    x->0

    rewrite as

    lim (1 + x[sin x/x])^(1/x)
    x->0
    use
    lim (1 + x*a)^(1/x)=exp(a)
    x->0
    and composition or squish theorem

    lim (e^(2 sin x) - e^(sin x)) / (sin 2x)
    x->0

    rewrite as
    lim (e^(2 y) - e^y) / y
    y->sin(x)->0

    and

    (e^(2 y) - e^y)=(e^y-1)^2+(e^y-1)

    and

    lim (e^x- 1) / x=1
    x->0
     
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