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Homework Help: Please help in limits

  1. Feb 20, 2006 #1
    hello everyone
    we've been doing some exercices of limits at class and there are many ones that i didn't understand... and since you don't get that chance to ask your teacher after class in Morcco, I came here for help!
    here are two of the tens that i didn't undersatnd:
    so to solve this limit:

    lim (x-1)/(sqrt(x²+1))
    x-> +infinity

    you have to go from or each x Є ]-infinity,0[ U ]0, +infinity[
    g(x)=(x-1)/(sqrt(x²+1))
    .........x(1-1/x)
    g(x)=-----------------
    .........|x| sqrt(1+1/x²)

    i just want to understand how you go from that first line to th second line !?

    and also on this one:

    how to go from
    [itex]sqrt(x^2+x+1)-x[/itex]
    to:
    [itex]x(sqrt(1+1/x+1/x^2)-1)) [/itex] for each x Є ]0,+infinity[

    i really appreciate your help, and also if there is a website that gives you the trucks to solve these kind of limits...thanks again
     
  2. jcsd
  3. Feb 20, 2006 #2

    Galileo

    User Avatar
    Science Advisor
    Homework Helper

    It's just bringing the x outside of the brackets. Isn't is clear that x(1-1/x) is the same as (x-1) for x/=0? Just expand the brackets.
    Same thing with sqrt(x^2+1). You can bring out the x^2 in (x^2+1), giving x^2(1+1/x^2) (valid for x/=0)

    But you don't need it to solve the limit. Intuitively you can argue that the -1 in the numerator and the +1 in the denominator are pretty insignificant for large x, so ignoring those you get x/|x|, whose limit is clear.
    You can also simply divide top and bottom of the fraction by x.
     
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