<s>[INST] Calculating the following limit: $\lim_{x \to 0} \frac{\sqrt{x^2+1}-\sqrt{x+1}}{1-\sqrt{x+1}} $ 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.
 [/INST] $$
\frac{\sqrt{x^2+1}-\sqrt{x+1}}{1-\sqrt{x+1}}=\frac{x^2-x}{-x}\cdot\frac{1+\sqrt{x+1}}{\sqrt{x^2+1}+\sqrt{x+1}}\sim1\cdot\frac{2}{2}
$$
</s>