# Introduction to limits

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

Moderate
Solution

## $\begin{array}{l}\underset{\mathrm{x}\to \mathrm{\infty }}{lim} \left(\sqrt[3]{{\mathrm{x}}^{3}+2{\mathrm{x}}^{2}}-\sqrt{{\mathrm{x}}^{2}+\mathrm{x}}\right)\\ =\underset{\mathrm{x}\to \mathrm{\infty }}{lim} \mathrm{x}\left[{\left(1+\frac{2}{\mathrm{x}}\right)}^{3}-{\left(1+\frac{1}{\mathrm{x}}\right)}^{\frac{1}{2}}\right]\\ =\underset{\mathrm{x}\to \mathrm{\infty }}{lim} \mathrm{x}\left[\left(1+\frac{1}{3}\cdot \frac{2}{\mathrm{x}}+\dots \right)-\left(1+\frac{1}{2}\cdot \frac{1}{\mathrm{x}}+\dots \right)\right]\\ =\underset{\mathrm{x}\to \mathrm{\infty }}{lim} \mathrm{x}\left(\frac{2}{3\mathrm{x}}-\frac{1}{2\mathrm{x}}\right)\\ =\frac{1}{6}\end{array}$

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The value of the limit $\underset{\mathrm{x}\to 0}{lim} {\left\{{1}^{1/{\mathrm{sin}}^{2}\mathrm{x}}+{2}^{1/{\mathrm{sin}}^{2}\mathrm{x}}+\dots +{\mathrm{n}}^{1/{\mathrm{sin}}^{2}\mathrm{x}}\right\}}^{{\mathrm{sin}}^{2}\mathrm{x}}$ is