The value of limn-n-1 2-n-1 2 - is

Lim_n→∞[√((n+1 )^2) √((n 1 )^2) ] =lim_n→∞n^23[(1+1n )^23 (1 1n )^23 ] Using binomial theorem expansions =lim_n→∞ n^23[43.1n+881.1n^3+..... ] =lim_n→∞[43·1n ...

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Standard XII

Mathematics

Right Hand Derivative

The value of ...

Question

The value of $lim n \to \infty [\sqrt[3]{{(n + 1)}^{2}} - \sqrt[3]{{(n - 1)}^{2}}]$ is

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Solution

$lim n \to \infty [\sqrt[3]{{(n + 1)}^{2}} - \sqrt[3]{{(n - 1)}^{2}}]$

$= lim n \to \infty n^{\frac{2}{3}} ⎡ ⎢ ⎢ ⎣ {(1 + \frac{1}{n})}^{\frac{2}{3}} - {(1 - \frac{1}{n})}^{\frac{2}{3}} ⎤ ⎥ ⎥ ⎦$
Using binomial theorem expansions
$= lim n \to \infty n^{\frac{2}{3}} [\frac{4}{3} . \frac{1}{n} + \frac{8}{81} . \frac{1}{n^{3}} + . . . . .]$
$= lim n \to \infty ⎡ ⎢ ⎢ ⎢ ⎣ \frac{4}{3} \cdot \frac{1}{n^{\frac{1}{3}}} + \frac{8}{81} \cdot \frac{1}{n^{\frac{7}{3}}} + . . . . ⎤ ⎥ ⎥ ⎥ ⎦ = 0$

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Right Hand Derivative

Standard XII Mathematics

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The value of limn→∞[3√(n+1)2−3√(n−1)2] is

limn→∞[3√(n+1)2−3√(n−1)2] =limn→∞n23⎡⎢ ⎢⎣(1+1n)23−(1−1n)23⎤⎥ ⎥⎦ Using binomial theorem expansions =limn→∞n23[43.1n+881.1n3+.....] =limn→∞⎡⎢ ⎢ ⎢⎣43⋅1n13+881⋅1n73+....⎤⎥ ⎥ ⎥⎦=0

The value of $lim n \to \infty [\sqrt[3]{{(n + 1)}^{2}} - \sqrt[3]{{(n - 1)}^{2}}]$ is