Tìm giới hạn : $I=\lim_{x\rightarrow 1}\frac{\sqrt[2013]{x}-\sqrt[2014]{x}}{\sqrt[2012]{x}-\sqrt[2013]{x}}$
$I=\lim_{x\rightarrow 1}\frac{\sqrt[2013]{x}-\sqrt[2014]{x}}{\sqrt[2012]{x}-\sqrt[2013]{x}}$
#1
Đã gửi 12-07-2015 - 20:34
#2
Đã gửi 13-07-2015 - 20:53
Đặt ${t^{2012.2013.2014}} = x $
$\mathop {\lim }\limits_{t \to 1} \frac{{{t^{2012.2014}} - {t^{2012.2013}}}}{{{t^{2013.2014}} - {t^{2012.2014}}}} = \mathop {\lim }\limits_{t \to 1} \frac{{{{\left( {{t^{2012}}} \right)}^{2014}} - {{\left( {{t^{2012}}} \right)}^{2013}}}}{{{{\left( {{t^{2014}}} \right)}^{2013}} - {{\left( {{t^{2014}}} \right)}^{2012}}}}$
$= \mathop {\lim }\limits_{t \to 1} \frac{{{{\left( {{t^{2012}}} \right)}^{2013}}\left( {{t^{2012}} - 1} \right)}}{{{{\left( {{t^{2014}}} \right)}^{2012}}\left( {{t^{2014}} - 1} \right)}} = \mathop {\lim }\limits_{t \to 1} \frac{{{{\left( {{t^{2012}}} \right)}^{2013}}\left( {{t^{2011}} + ... + 1} \right)}}{{{{\left( {{t^{2014}}} \right)}^{2012}}\left( {{t^{2013}} + ... + 1} \right)}}$
$= \frac{{2012}}{{2014}}$
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