\(lim\left(\sqrt{4n^2+n}-\sqrt{4n^2+2}\right)\)
\(=lim\dfrac{\left(\sqrt{4n^2+n}-\sqrt{4n^2+2}\right)\times\left(\sqrt{4n^2+n}+\sqrt{4n^2+2}\right)}{\left(\sqrt{4n^2+n}+\sqrt{4n^2+2}\right)}\)
\(=lim\dfrac{\left(\sqrt{4n^2+n}\right)^2-\left(\sqrt{4n^2+2}\right)^2}{\sqrt{4n^2+n}+\sqrt{4n^2+2}}\)
\(=lim\dfrac{4n^2+n-4n^2-2}{\sqrt{4n^2+n}+\sqrt{4n^2+2}}\)
\(=lim\dfrac{n-2}{\sqrt{4n^2+n}+\sqrt{4n^2+2}}\)
\(=lim\dfrac{\dfrac{n}{n}-\dfrac{2}{n}}{\dfrac{n}{n}\sqrt{\dfrac{4n^2}{n^2}+\dfrac{n}{n^2}}+\dfrac{n}{n}\sqrt{\dfrac{4n^2}{n^2}+\dfrac{2}{n^2}}}\)
\(=\dfrac{1-0}{1\sqrt{4+0}+1\sqrt{4+0}}\)
\(=\dfrac{1}{2+2}\)
\(=\dfrac{1}{4}\)
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