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How do you solve the following limit without a calculator?

  1. Nov 27, 2004 #1
    Hi,

    I was wondering, how would one solve the following equation without using a calculator. In other words, algebraically.

    lim (x + sqrt(x^2+5x))
    x-> -infinity

    Thanks in advance
     
  2. jcsd
  3. Nov 27, 2004 #2

    arildno

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    Multiply with the conjugate expression:
    [tex]x+\sqrt{x^{2}+5x}=(x+\sqrt{x^{2}+5x})\frac{x-\sqrt{x^{2}+5x}}{x-\sqrt{x^{2}+5x}}=-\frac{5x}{x-\sqrt{x^{2}+5x}}\to-\frac{5}{2}, x\to\infty[/tex]
     
  4. Nov 27, 2004 #3
    Thanks a lot. Really appreciate it. I thought you had to do it a certain way because the limit is approaching infinity instead of a number.
     
  5. Nov 27, 2004 #4
    Didn't notice this, but how does the bottom become 2?
     
  6. Nov 27, 2004 #5

    Because when calculating the limit to -infinity you need to put the denominator [tex]x-\sqrt{x^{2}+5x}[/tex] in factorized form. When doing so you need to get an x² out of the square-root but realize that x is negative so you need to write [tex]x-(-x)\sqrt{1+\frac{5x}{x^2}}[/tex]. This is just like saying that [tex]\sqrt{9} = \sqrt{(-3)(-3)} = -3[/tex]. Factoring on you will get that [tex]x(1+\sqrt{1+\frac{5}{x}})[/tex] and the x will vanish because of the x you will get in the nominator after completing the exact same procedure there. If you fill in [tex]- \infty[/tex] you will get the 2 in the bottom


    regards
    marlon
     
  7. Nov 27, 2004 #6
    Thank you!
     
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