Evaluate $\displaystyle \large \lim_{x \,\to\, \infty}{\normalsize (\sqrt{x^2+2x}-x)}$

Let us try to evaluate the limit of the algebraic function $\sqrt{x^2+2x}-x$ as $x$ approaches infinity by the direct substitution method.

$\displaystyle \large \lim_{x \,\to\, \infty}{\normalsize (\sqrt{x^2+2x}-x)}$

$= \,\,\,$ $\sqrt{{(\infty)}^2+2(\infty)}-(\infty)$

$= \,\,\, \infty$

The limit of algebraic function as $x$ approaches infinity is undefined. So, try to find the limit of the function in another method.

Rationalize the function

The algebraic function is in radical form and the limit of this function is undefined as $x$ approaches infinity. So, the limit of this radical function can be calculated by using rationalization method by multiplying and dividing the function by its conjugate function.

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, \infty}{\normalsize \Bigg[(\sqrt{x^2+2x}-x) \times 1 \Bigg]}$

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, \infty}{\normalsize (\sqrt{x^2+2x}-x)}$ $\times$ $\dfrac{\sqrt{x^2+2x}+x}{\sqrt{x^2+2x}+x}$

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, \infty}{\normalsize \dfrac{(\sqrt{x^2+2x}-x) \times (\sqrt{x^2+2x}+x)}{\sqrt{x^2+2x}+x}}$

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, \infty}{\normalsize \dfrac{{(\sqrt{x^2+2x})}^2-x^2}{\sqrt{x^2+2x}+x}}$

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, \infty}{\normalsize \dfrac{x^2+2x-x^2}{\sqrt{x^2+2x}+x}}$

$= \,\,\,$ $\require{cancel} \displaystyle \large \lim_{x \,\to\, \infty}{\normalsize \dfrac{\cancel{x^2}+2x-\cancel{x^2}}{\sqrt{x^2+2x}+x}}$

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, \infty}{\normalsize \dfrac{2x}{\sqrt{x^2+2x}+x}}$

Simplify the function

$x$ is a factor in the numerator and take $x$ common from all the terms of the denominator for making them to get cancelled.

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, \infty}{\normalsize \dfrac{2x}{\sqrt{x^2 \Big(1+2 \times \dfrac{1}{x}\Big)}+x}}$

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, \infty}{\normalsize \dfrac{2x}{x\sqrt{1+\dfrac{2}{x}}+x}}$

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, \infty}{\normalsize \dfrac{2x}{x\Bigg[\sqrt{1+\dfrac{2}{x}}+1\Bigg]}}$

$= \,\,\,$ $\require{cancel} \displaystyle \large \lim_{x \,\to\, \infty}{\normalsize \dfrac{2\cancel{x}}{\cancel{x}\Bigg[\sqrt{1+\dfrac{2}{x}}+1\Bigg]}}$

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, \infty}{\normalsize \dfrac{2}{\sqrt{1+\dfrac{2}{x}}+1}}$