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Algebra of Limits

Find the limi...

Question

Find the limit $lim x \to \infty$ $((x + 1)^{\frac{2}{3}} - (x - 1)^{\frac{2}{3}})$

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Solution

$l e t x - 1 = t a s x \to \infty, t \to \infty {lim}_{t \to \infty} (t + 2)^{(\frac{2}{3})} - (t)^{(\frac{2}{3})} = {lim}_{t \to \infty} (\frac{(t + 2)}{t})^{(\frac{2}{3})} - 1 \cdot (t)^{(\frac{2}{3})} = {lim}_{t \to \infty} (\frac{(1 + (\frac{2}{t}))^{(\frac{2}{3})}}{(\frac{2}{t})}) \cdot (\frac{2}{t}) \cdot t^{(\frac{2}{3})} = (\frac{2}{3}) \cdot (\frac{2}{\infty}) = 0$

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