Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

How do you solve the following limit without a calculator?

  1. Nov 27, 2004 #1

    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


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    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

  7. Nov 27, 2004 #6
    Thank you!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook