Question-173499

Question Number 173499 by cortano1 last updated on 12/Jul/22

Commented by blackmamba last updated on 12/Jul/22

x = 1 + r

L = \lim_{r \to 0} \frac{\sqrt[3]{7 + {(1 + r)}^{3}} - \sqrt{3 + {(1 + r)}^{2}}}{r}

= \lim_{r \to 0} \frac{\sqrt[3]{8 + (3 r + 3 r^{2} + r^{3})} - \sqrt{4 + (2 r + r^{2})}}{r}

= \lim_{r \to 0} \frac{2 \sqrt[3]{1 + (\frac{3 r + 3 r^{2} + r^{3}}{8})} - 2 \sqrt{1 + (\frac{2 r + r^{2}}{4})}}{r}

= \lim_{r \to 0} \frac{2 (1 + \frac{3 r + 3 r^{2} + r^{3}}{24}) - 2 (1 + (\frac{2 r + r^{2}}{8}))}{r}

= \lim_{r \to 0} \frac{3 + 3 r + r^{2}}{12} - \frac{2 + r}{4}

= - \frac{1}{4}

Commented by Tawa11 last updated on 13/Jul/22

Great sir

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