Difference of roots

Statement

Calculate $\displaystyle\lim_{x\to+\infty}\bigl(\sqrt{x^2+3x}-\sqrt{x^2-x}\bigr)$.

Solution

An $\infty-\infty$ form: we rationalise by multiplying and dividing by the conjugate $\sqrt{x^2+3x}+\sqrt{x^2-x}$: $= \lim_{x\to+\infty}\frac{(x^2+3x)-(x^2-x)}{\sqrt{x^2+3x}+\sqrt{x^2-x}} = \lim_{x\to+\infty}\frac{4x}{\sqrt{x^2+3x}+\sqrt{x^2-x}}.$ We factor out $x$ (for $x\to+\infty$, $x>0$ and $\sqrt{x^2}=x$): $= \lim_{x\to+\infty}\frac{4x}{x\left(\sqrt{1+\frac{3}{x}}+\sqrt{1-\frac{1}{x}}\right)} = \frac{4}{1+1} = \boxed{2}.$

Links

Topics: Limits
Concepts: Indeterminate forms
Skills: Calculating limits
Exercise type: Limit calculation