how to find the limit of $\lim_{x \to0}\frac{\sqrt{x^2+x+1}-\sqrt{x+1}}{x^2}$

Question

How to find the limit of $$\lim_{x \to0}\frac{\sqrt{x^2+x+1}-\sqrt{x+1}}{x^2}\,?$$ I tried L'Hospital's rule, but it didn't work well.

Can I have some assistance? Thank you in advance

Answer 1

Multiply numerator and denominator by $$\sqrt{x^2+x+1}+\sqrt{x+1}.$$ You will get $$\frac{1}{\sqrt{x^2+x+1}+\sqrt{x+1}}$$ and the limit is $$\frac{1}{2}$$

Answer 2

Since, near $0$,$$\sqrt{x^2+x+1}=1+\frac x2+\frac{3x^2}8+O(x^3)$$and$$\sqrt{x+1}=1+\frac x2-\frac{x^2}8+O(x^3),$$then$$\lim_{x\to0}\frac{\sqrt{x^2+x+1}-\sqrt{x+1}}{x^2}=\frac38-\left(-\frac18\right)=\frac12.$$

Answer 3

why apply l hopital when you can rationlaise..

$$\lim_{x \to0}\frac{\sqrt{x^2+x+1}-\sqrt{x+1}}{x^2}$$

and you will get the following

$$\lim_{x \to0}\frac{1}{\sqrt{x^2+x+1}+\sqrt{x+1}}$$

put $ x =0 $ you will get $\frac{1}{2}$