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[TomaszSy]

a) \(\Lim_{x\to \infty } ( \sqrt[3]{(x+1)^2} - \sqrt[3]{(x-1)^2} )\)
b) \(\Lim_{x\to \infty } x^{ \frac{3}{2} } ( \sqrt{x^3+1} - \sqrt{x^3-1}) \)
c) \(\Lim_{n\to \infty } ( \frac{1}{1*3} + \frac{1}{3*5} + \ldots + \frac{1}{(2n-1)(2n+1)} )\)

[eresh]

a) \(\Lim_{x\to \infty } ( \sqrt[3]{(x+1)^2} - \sqrt[3]{(x-1)^2} )\)

\(\Lim_{x\to \infty}\frac{(x+1)^2-(x-1)^2}{(x+1)\sqrt[3]{x+1}+\sqrt[3]{(x+1)^2(x-1)^2}+(x-1)\sqrt[3]{x-1}}=\\
\Lim_{x\to \infty}\frac{4x}{x((1+\frac{1}{x})\sqrt[3]{x+1}+\sqrt[3]{x-\frac{2}{x}+\frac{1}{x^3}}+(1-\frac{1}{x})\sqrt[3]{x-1})}=\\
\Lim_{x\to\infty}\frac{4}{(1+\frac{1}{x})\sqrt[3]{x+1}+\sqrt[3]{x-\frac{2}{x}+\frac{1}{x^3}}+(1-\frac{1}{x})\sqrt[3]{x-1}}=[\frac{4}{\infty}]=0
\)

[eresh]

b) \(\Lim_{x\to \infty } x^{ \frac{3}{2} } ( \sqrt{x^3+1} - \sqrt{x^3-1}) \)

\(\Lim_{x\to \infty } x^{ \frac{3}{2} } ( \sqrt{x^3+1} - \sqrt{x^3-1}) =\\
\Lim_{x\to\infty}\frac{\sqrt{x^3}(x^3+1-x^3+1)}{\sqrt{x^3+1} + \sqrt{x^3-1}}=\\
=\Lim_{x\to\infty}\frac{2\sqrt{x^3}}{\sqrt{x^3}(\sqrt{1+\frac{1}{x^3}}+\sqrt{1-\frac{1}{x^3}})}=\\
=\Lim_{x\to\infty}\frac{2}{(\sqrt{1+\frac{1}{x^3}}+\sqrt{1-\frac{1}{x^3}})}=\frac{2}{2}=1\)

[eresh]

c) \(\Lim_{n\to \infty } ( \frac{1}{1*3} + \frac{1}{3*5} + \ldots + \frac{1}{(2n-1)(2n+1)} )\)

\(\Lim_{n\to \infty } ( \frac{1}{1*3} + \frac{1}{3*5} + \ldots + \frac{1}{(2n-1)(2n+1)} )=\\
=\Lim_{n\to\infty}(\frac{1}{2}-\frac{1}{6}+\frac{1}{6}-\frac{1}{10}+...+\frac{0,5}{2n-1}-\frac{0,5}{2n+1})=\\
=\Lim_{n\to\infty}(\frac{1}{2}-\frac{0,5}{2n+1)}=0,5-0=0,5\)