Calculus

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Limit as x goes to negative infinity of sqrt(x^2+3x)+x?

  • Calculus -

    It depends upon whether you take the positive or negative square root. If you take the positive root and x-> -infinity, the first term approaches = |x| = -x (for negative x). Adding x to that gives you zero in the limit.

    Check, let x = -1000
    sqrt(10^6 -3000)= 999.5
    Subtract -1000 and you get -0.5
    Let x = -10,000
    sqrt(10^8- 3^10^4) = 999.85
    Suntract -10,000 and you get -0.15

  • Calculus -

    I still don't get it. The answer is -1.5, how do I get that w/o a calculator?

  • Calculus -

    You are correct, both my derivation and my numerical examples were wrong. You can get -1.5 by considering a series expansion of sqrt(1+y) = 1 + y/2 + ...

    sqrt(x^2 + 3x) = sqrt[x^2*(1 + (3/x)]
    = x * [1 + (3/2)x + higher order terms in 3/x] = x + 3/2 + ...
    As x approaches -infinity, with positive square roots only taken, the first term sqrt(x^2) approaches |x| and the second term remains -3/2
    Adding x to that is like subtracting
    -|x|, leaving you with -3/2

    Fox x = -1000, f(x) = -1.5011
    for x = -10000, f(x) = -1.5001

    I apologize for my sloppy math.

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