Question 54 of 90
Q). $\underset{x\to \infty }{\lim }\sqrt{x^2+x+1}-\sqrt{x^2-x-1}$
1.  1
2.  -2
3.  2/3
4.  -2/3

Dear Student,
Please find below the solution to the asked query:

$$\begin{gathered} \underset{x\to \infty }{\lim } \sqrt{x^2+x+1}-\sqrt{x^2-x-1} \\ =\underset{x\to \infty }{\lim } \left(\sqrt{x^2+x+1}-\sqrt{x^2-x-1}\right)\left[\frac{\sqrt{x^2+x+1}+\sqrt{x^2-x-1}}{\sqrt{x^2+x+1}+\sqrt{x^2-x-1}}\right] \\ =\underset{x\to \infty }{\lim } \frac{x^2+x+1-x^2+x+1}{\sqrt{x^2+x+1}+\sqrt{x^2-x-1}} \\ =\underset{x\to \infty }{\lim } \frac{2x+2}{\sqrt{x^2+x+1}+\sqrt{x^2-x-1}} \\ =\frac{2}{1+1} \\ =\frac{2}{2} \\ =1 \end{gathered}$$

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