$$\lim_{x\to 0} \dfrac{\sqrt{1-x}-\sqrt{1+x}}{x^2-3x}$$

I am stuck at radicals. Division by 1/x doesn't help.

  • $\begingroup$ I formatted your post. Tell me if this was not the intended meaning. $\endgroup$ Oct 17, 2013 at 5:58
  • $\begingroup$ The Maple command $$ Student[Calculus1]:-LimitTutor((sqrt(1-x)-sqrt(1+x))/(x^2-3*x), x = 0) $$ finds it step by step with explanation. See that link for info. BTW, I have strong reasons to think that some users of MathematicsSE have many accounts and vote up themselves. $\endgroup$
    – user64494
    Oct 17, 2013 at 7:16
  • $\begingroup$ The Maple commands won't do your exams for you and, even worse, won't let you get into the marvelous world of thinking and reasoning to get (closer, at least) at some problem's solution. $\endgroup$
    – DonAntonio
    Oct 17, 2013 at 10:57

1 Answer 1



Rationalize the numerator $$\sqrt{1-x}-\sqrt{1+x}=\frac{(1-x)-(1+x)}{\sqrt{1-x}+\sqrt{1+x}}=\frac{-2x}{\sqrt{1-x}+\sqrt{1+x}}$$

Then cancel out $x$ form the numerator & denominator as $x\ne0$ as $x\to0$

  • $\begingroup$ I am getting radicals from -1 when dividing x nom and denom $\endgroup$
    – J.Olufsen
    Oct 17, 2013 at 6:14
  • $\begingroup$ @RCola, we get $$\frac{-2x}{\sqrt{1-x}+\sqrt{1+x}}\cdot\frac1{x(x-3)}=\frac{-2}{(\sqrt{1-x}+\sqrt{1+x})(x-3)}$$ Put $x=0$ $\endgroup$ Oct 17, 2013 at 6:17

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