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Understanding the algebra behind these limit problems

  1. Nov 3, 2016 #1
    1. The problem statement, all variables and given/known data
    $$\lim_{x \to -∞}{\sqrt{x^2 + bx + c} - x}.$$


    2. Relevant equations


    3. The attempt at a solution
    So in problem 1, once I got to a point where I am to divide by the highest power in the denominator(x) I get something like:
    $$\lim_{x \to -∞}\frac{bx+c}{\sqrt{x^2+bx+c}+x}$$

    Now what I want to clarify for myself is if the following is correct:
    $$\lim_{x \to -∞}\frac{bx+c}{\sqrt{x^2}\sqrt{1+b/x+c/x^2}+x}$$
    $$\lim_{x \to -∞}\frac{b+c/x}{(|x|/x)\sqrt{1+b/x+c/x^2}+1)}$$
    $$\lim_{x \to -∞}\frac{b+0}{(x)/(x)\sqrt{1+0+0}+1}$$
    and you get $$b/2$$

    So, when you have $$\frac{|x|}{(x)}$$ how would I go about dividing it in this situation? I know that if we are looking at a left sided limit of say 0, |x| = -(x) and vice versa for the right sided limit.

    But this limit is at negative infinity, so how would it work in this situation? Please let me know if I'm wrong in my algebra approach here.
     
    Last edited: Nov 3, 2016
  2. jcsd
  3. Nov 3, 2016 #2

    Math_QED

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    Homework Helper

    ##\sqrt{x^2}= -x## when ##x \rightarrow -\infty##
     
  4. Nov 3, 2016 #3
    Wouldn't that give me b/0?
     
  5. Nov 3, 2016 #4

    Mark44

    Staff: Mentor

    Try multiplying by ##\sqrt{x^2 - x} - x## over itself (i.e., multiplying by 1).

    Edit: Original problem was changed, so this advice no longer applies.
     
    Last edited: Nov 3, 2016
  6. Nov 3, 2016 #5
    Jesus I had just realized I wrote down the wrong problem. Going to change it
     
  7. Nov 3, 2016 #6
    Okay... so at this point is where I'm confused if I'm wrong/right:

    $$\frac{b+c/x}{\sqrt{x^2}\sqrt{1+b/x+c/x^2} + x}$$


    $$\frac{b+c/x}{{|x|/x}\sqrt{1+b/x+c/x^2} + x}$$

    $$\frac{b+0}{{-(x)/x}\sqrt{1+0+0} + x}$$

    and then I get b/0 which is not the correct answer. Where did I go wrong here?
     
  8. Nov 3, 2016 #7

    Mark44

    Staff: Mentor

    Is this still the problem?

    $$\lim_{x \to -∞}{\sqrt{x^2 + bx + c} - x}.$$
    If so, you can evaluate it almost directly -- you don't need to multiply by the conjugate over itself, which was my earlier hint.
     
  9. Nov 3, 2016 #8
    So basically multiply it by $$\frac{\sqrt{x^2+bx+c}- x}{\sqrt{x^2+bx+c}- x}$$ ?
    $$\lim_{x\to -∞}\frac{2x^2+bx+c}{\sqrt{x^2+bx+c}-x}$$
    Then I divided by the highest power in the denominator(x) and evaluated it at -∞ to get this:

    $$\lim_{x \to -∞}\frac{2(-∞)+b}{\sqrt{1 +0 + 0} - 1}.$$ Is what I got, which ended up being -∞/0. Where am I messing up? The answer is b/2(which I coincidentally got in the OP but it was done incorrectly)
     
  10. Nov 3, 2016 #9

    Mark44

    Staff: Mentor

    What I said was "you DON'T need to multiply by the conjugate over itself."
    Also, the work just above isn't kosher -- you don't plug in ##\infty##.
     
  11. Nov 3, 2016 #10
    $$\frac{\sqrt{x^2+bx+c}- x}{\sqrt{x^2+bx+c}- x}$$ Is not the conjugate

    I thought you had meant to multiply both the numerator and the denominator by itself(because this evaluates to 1). Could you further elaborate on what you meant?
     
  12. Nov 3, 2016 #11

    Ray Vickson

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    $$\lim_{x \to -\infty} \left[ \sqrt{x^2+bx+c}-x \right] = \lim_{t \to +\infty} \left[ \sqrt{t^2 - bt +c} + t \right],$$
    by taking ##x = -t##. Note that ##\sqrt{t^2 - b t + c} = \sqrt{t^2} \sqrt{1-z} = t \sqrt{1-z}##, where ##z = (bt-c)/t^2 \to 0## as ##t \to \infty##. (Note that for ##t > 0## we have ##\sqrt{t^2} = t##.)
     
  13. Nov 3, 2016 #12

    Mark44

    Staff: Mentor

    Right, it's not the conjugate over itself, but what's the point of doing this? My comment about multiplying by the conjugate over itself (which is 1) applied to the original problem, which you changed.
     
  14. Nov 4, 2016 #13
    My prof stated that this question was supposed to be the limit at infinity, not negative infinity.

    My approach was correct in the original post however I put $$|x|/x$$ = $$(x)/x$$ at negative infinity which was my mistake.
     
  15. Nov 4, 2016 #14
    Is there a thereom/name for what you're doing here? I've never seen this before

    What were you suggesting with the other problem? Just for reference
     
  16. Nov 4, 2016 #15

    Mark44

    Staff: Mentor

    AFAIK, this doesn't rise to the level of needing a theorem to justify it. All that's going on is substitution.

    If you are working with a limit in the form of ##\frac{a + b}c##, and both numerator and denominator evaluate to 0 (or both are infinite), it can sometimes be helpful to multiply by 1 in the form of the conjugate over itself. I.e., work with ##\frac{a + b}c \cdot \frac{a - b}{a - b} = \frac{a^2 - b^2}{c(a - b)}##.
     
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