limn→∞(n+1)45−(n−1)45=

limn→∞(n+1)45−(n−1)45=limn→∞n4/51+1n4/5−1−1n4/5=limn→∞n4/51+451n+4545−12!1n2+...−1−451n+4545−12!1n2+...=limn→∞n4/5851n+41251n3+...=limn→∞851n1/5+41251n12/5+...=0

limn→∞(n+1)45−(n−1)45=

limn→∞(n+1)45−(n−1)45=limn→∞n4/51+1n4/5−1−1n4/5=limn→∞n4/51+451n+4545−12!1n2+...−1−451n+4545−12!1n2+...=limn→∞n4/5851n+41251n3+...=limn→∞851n1/5+41251n12/5+...=0

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$\lim_{n \to \infty} (\sqrt[5]{{(n + 1)}^{4}} - \sqrt[5]{{(n - 1)}^{4}}) =$

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Detailed Solution

$\begin{array}{l} \lim_{n \to \infty} (\sqrt[5]{{(n + 1)}^{4}} - \sqrt[5]{{(n - 1)}^{4}}) = \lim_{n \to \infty} n^{4 / 5} ({(1 + \frac{1}{n})}^{4 / 5} - {(1 - \frac{1}{n})}^{4 / 5}) \\ = \lim_{n \to \infty} n^{4 / 5} ((1 + \frac{4}{5} \frac{1}{n} + \frac{\frac{4}{5} (\frac{4}{5} - 1)}{2!} \frac{1}{n^{2}} + ...) - (1 - \frac{4}{5} \frac{1}{n} + \frac{\frac{4}{5} (\frac{4}{5} - 1)}{2!} \frac{1}{n^{2}} + ...)) \\ = \lim_{n \to \infty} n^{4 / 5} (\frac{8}{5} \frac{1}{n} + \frac{4}{125} \frac{1}{n^{3}} + ...) \\ = \lim_{n \to \infty} (\frac{8}{5} \frac{1}{n^{1 / 5}} + \frac{4}{125} \frac{1}{n^{12 / 5}} + ...) \\ = 0 \end{array}$