Chủ đề này có 1 trả lời

[naruto01]

Tìm giới hạn : $I=\lim_{x\rightarrow 1}\frac{\sqrt[2013]{x}-\sqrt[2014]{x}}{\sqrt[2012]{x}-\sqrt[2013]{x}}$

[nguyenlyninhkhang]

Đặt  ${t^{2012.2013.2014}} = x $

$\mathop {\lim }\limits_{t \to 1} \frac{{{t^{2012.2014}} - {t^{2012.2013}}}}{{{t^{2013.2014}} - {t^{2012.2014}}}} = \mathop {\lim }\limits_{t \to 1} \frac{{{{\left( {{t^{2012}}} \right)}^{2014}} - {{\left( {{t^{2012}}} \right)}^{2013}}}}{{{{\left( {{t^{2014}}} \right)}^{2013}} - {{\left( {{t^{2014}}} \right)}^{2012}}}}$

$= \mathop {\lim }\limits_{t \to 1} \frac{{{{\left( {{t^{2012}}} \right)}^{2013}}\left( {{t^{2012}} - 1} \right)}}{{{{\left( {{t^{2014}}} \right)}^{2012}}\left( {{t^{2014}} - 1} \right)}} = \mathop {\lim }\limits_{t \to 1} \frac{{{{\left( {{t^{2012}}} \right)}^{2013}}\left( {{t^{2011}} + ... + 1} \right)}}{{{{\left( {{t^{2014}}} \right)}^{2012}}\left( {{t^{2013}} + ... + 1} \right)}}$

$= \frac{{2012}}{{2014}}$