If $$L=limx$$→∞ $$x3+2x23$$−$$x2+x$$ then the value of $$3/L$$ is $$______$$ .

limx→∞ $$x3+2x23$$−$$x2+x=limx$$→∞ $$x1+2x3$$−$$1+1x12=limx$$→∞ $$x1+13$$⋅$$2x+$$…−$$1+12$$⋅$$1x+$$…$$=limx$$→∞ $$x23x$$−$$12x=16$$

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If $L = lim_{x \to \infty} (\sqrt[3]{x^{3} + 2 x^{2}} - \sqrt{x^{2} + x})$ then the value of 3/L is ______ .

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Solution

$\begin{array}{l} lim_{x \to \infty} (\sqrt[3]{x^{3} + 2 x^{2}} - \sqrt{x^{2} + x}) \\ = lim_{x \to \infty} x [{(1 + \frac{2}{x})}^{3} - {(1 + \frac{1}{x})}^{\frac{1}{2}}] \\ = lim_{x \to \infty} x [(1 + \frac{1}{3} \cdot \frac{2}{x} + \dots) - (1 + \frac{1}{2} \cdot \frac{1}{x} + \dots)] \\ = lim_{x \to \infty} x (\frac{2}{3 x} - \frac{1}{2 x}) \\ = \frac{1}{6} \end{array}$

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Introduction to limits

Remember concepts with our Masterclasses.

If L=limx→∞ x3+2x23−x2+x then the value of 3/L is ______ .

limx→∞ x3+2x23−x2+x=limx→∞ x1+2x3−1+1x12=limx→∞ x1+13⋅2x+…−1+12⋅1x+…=limx→∞ x23x−12x=16

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If $L = lim_{x \to \infty} (\sqrt[3]{x^{3} + 2 x^{2}} - \sqrt{x^{2} + x})$ then the value of 3/L is ______ .