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lim (x--> 2) \(\frac{\sqrt{2x+5}-\sqrt{7+x}}{x^2-2x}\)

\(_{x\underrightarrow{lim}2}\frac{\sqrt{2x+5}-\sqrt{7-x}}{x^2-2x}\)

\(=x\underrightarrow{lim}2\frac{\left(\sqrt{2x+5}-\sqrt{7+x}\right)\left(\sqrt{2x+5}+\sqrt{7+x}\right)}{\left(x^2-2x\right)\left(\sqrt{2x+5}+\sqrt{7+x}\right)}\)

\(=x\underrightarrow{lim}2\frac{1}{x\left(\sqrt{2x+5}+\sqrt{7+x}\right)}=\frac{1}{12}\)

a)lim \(\frac{\sqrt{n^2-4n}-\sqrt{4n+1}}{\sqrt{3n^2+1}+n}\)

=lim \(\frac{\sqrt{1-\frac{4}{n}}-\sqrt{\frac{4}{n}+\frac{1}{n^2}}}{\sqrt{3+\frac{1}{n^2}}+1}=\frac{1}{\sqrt{3}+1}\)

b)lim  \(\frac{\sqrt[3]{8n^3+n^2}-n}{2n-3}\)

= lim \(\frac{\sqrt[3]{8+\frac{1}{n^3}}-1}{2-\frac{3}{n}}=\frac{2-1}{2}=\frac{1}{2}\)

lim\(\left(\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{n\left(n+1\right)}\right)\)

= lim \(\left(\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{n}-\frac{1}{n+1}\right)\)

= lim \(\left(\frac{1}{3}-\frac{1}{n+1}\right)\)

= 1/3

\(\text{GIẢI :}\)

\(lim\left(\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+...+\frac{1}{\text{n}\left(\text{n}+1\right)}\right)\)

\(=lim\left(\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{\text{n}}-\frac{1}{\text{n}+1}\right)\)

\(=lim\left(\frac{1}{3}-\frac{1}{\text{n}\left(\text{n + 1}\right)}\right)\)

\(=\frac{1}{3}\)