Evaluate: _ y| ( x^ 2 +x + 1 -x ). - Vidyadip

Question

Evaluate:
$\lim\limits_{\text{y|} \to \infty}\Big(\sqrt{\text{x}^{2}+\text{x + 1}}-\text{x}\Big).$

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Answer

$\lim\limits_{x \to \infty}\Big(\sqrt{\text{x}^{2}+\text{x + 1}}-\text{x}\Big)=\lim\limits_{\text{x}\rightarrow\infty}\frac{\Big(\sqrt{\text{x}^{2}+\text{x + 1}}-\text{x}\Big)\Big(\sqrt{\text{x}^{2}+\text{x + 1}}+\text{x}\Big)}{\Big(\sqrt{\text{x}^{2}+\text{x + 1}}+\text{x}\Big)}$
$=\lim\limits_{\text{x}\rightarrow\infty}\frac{\text{x + 1}}{\sqrt{\text{x}^{2}+\text{x + 1}}+\text{x}}=\lim\limits_{\text{x}\rightarrow\infty}\frac{1+\frac{1}{\text{x}}}{\sqrt{1+\frac{1}{\text{x}}+\frac{1}{\text{x}^{2}}+1}}$
$\frac{1}{1+1}=\frac{1}{2}.$

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