Find ${\lim }_{x\to \infty }(\sqrt{x^2+1}-x)$.

Solution

1. Multiply and divide by the conjugate: $\frac{(\sqrt{x^2+1}-x)(\sqrt{x^2+1}+x)}{\sqrt{x^2+1}+x}$.
2. Simplify the numerator: $\frac{x^2+1-x^2}{\sqrt{x^2+1}+x}=\frac{1}{\sqrt{x^2+1}+x}$.
3. As $x\to \infty$, the denominator approaches $\infty$.
4. The limit is $1/\infty =0$.
5. Hence the answer is (A).