Rationalisation

When roots appear in an $\infty-\infty$ form, multiplying and dividing by the conjugate eliminates the difference of roots and makes the form tractable.

Example — Rationalisation ( $\infty-\infty$ form)

$\lim_{x\to+\infty}\bigl(\sqrt{x^2+x}-x\bigr).$ We multiply and divide by the conjugate $\sqrt{x^2+x}+x$: $= \lim\frac{(x^2+x)-x^2}{\sqrt{x^2+x}+x} = \lim\frac{x}{\sqrt{x^2+x}+x} = \lim\frac{x}{x\left(\sqrt{1+1/x}+1\right)} = \frac{1}{1+1} = \frac{1}{2}.$

After rationalising, by factoring out $x$ (for $x\to+\infty$) under the root and in the denominator, the indeterminate form resolves into a finite value.

Links

Topics: Limits
Concepts: Indeterminate forms
Skills: Calculating limits