Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am trying to calculate this limit: $$ \lim_{x \to 0} \frac{\sqrt{x^2+1}-\sqrt{x+1}}{1-\sqrt{x+1}} $$

I've tried using conjugate of both denominator and numerator but I can't get the right result.

share|cite|improve this question
up vote 4 down vote accepted

$$\frac{\sqrt{x^2+1}-\sqrt{x+1}}{1-\sqrt{x+1}}$$ $$=\frac{(1+\sqrt{x+1})\{x^2+1-(x+1)\}}{(\sqrt{x^2+1}+\sqrt{x+1})(1-(x+1))}$$ $$=\frac{(1+\sqrt{x+1})\{x(x-1)\}}{(\sqrt{x^2+1}+\sqrt{x+1})(-x)}$$ $$=\frac{(1+\sqrt{x+1})(1-x)}{(\sqrt{x^2+1}+\sqrt{x+1})}\text { if } x\ne0$$

As $x\to0,x\ne0$

So, $$\lim_{x\to0}\frac{\sqrt{x^2+1}-\sqrt{x+1}}{1-\sqrt{x+1}}=\lim_{x\to0}\frac{(1+\sqrt{x+1})(1-x)}{(\sqrt{x^2+1}+\sqrt{x+1})}=\frac{(1+1)}{(1+1)}=1$$

Alternatively, as $\lim_{x\to0}\frac{\sqrt{x^2+1}-\sqrt{x+1}}{1-\sqrt{x+1}}$ is of the form $\frac00,$

Applying L'Hospital's Rule we get, $$\lim_{x\to0}\frac{\sqrt{x^2+1}-\sqrt{x+1}}{1-\sqrt{x+1}}$$ $$=\lim_{x\to0}\frac{\frac x{\sqrt{x^2+1}}-\frac1{2\sqrt{x+1}}}{-\frac1{2\sqrt{x+1}}}$$ $$=\lim_{x\to0}\left(1-\frac{2x\sqrt{x+1}}{\sqrt{x^2+1}}\right)\text{ as }x+1\ne0$$ $$=1$$

share|cite|improve this answer

$$ \frac{\sqrt{x^2+1}-\sqrt{x+1}}{1-\sqrt{x+1}}=\frac{x^2-x}{-x}\cdot\frac{1+\sqrt{x+1}}{\sqrt{x^2+1}+\sqrt{x+1}}\sim1\cdot\frac{2}{2} $$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.