<s>[INST] Please help me with this limit $$
\lim_{h\to 0} {\frac {\sqrt[3]{x+h}-\sqrt[3]{x}} h}
$$
I tried with the conjugation in the top and under but i still have the determiantion the result must be 1/3 i did it with Matlab, you CAN'T use Lhopital
 [/INST] Just multiply both side to $ \left( \sqrt [ 3 ]{ { \left( x+h \right)  }^{ 2 } } +\sqrt [ 3 ]{ x\left( x+h \right)  } +\sqrt [ 3 ]{ { x }^{ 2 } }  \right) $ to get announced result $$\lim _{ h\to 0 }{ \frac { \sqrt [ 3 ]{ x+h } -\sqrt [ 3 ]{ x }  }{ h }  } =\lim _{ h\to 0 }{ \frac { \left( \sqrt [ 3 ]{ x+h } -\sqrt [ 3 ]{ x }  \right) \left( \sqrt [ 3 ]{ { \left( x+h \right)  }^{ 2 } } +\sqrt [ 3 ]{ x\left( x+h \right)  } +\sqrt [ 3 ]{ { x }^{ 2 } }  \right)  }{ h\left( \sqrt [ 3 ]{ { \left( x+h \right)  }^{ 2 } } +\sqrt [ 3 ]{ x\left( x+h \right)  } +\sqrt [ 3 ]{ { x }^{ 2 } }  \right)  }  } =\\ =\lim _{ h\to 0 }{ \frac { h }{ h\left( \sqrt [ 3 ]{ { \left( x+h \right)  }^{ 2 } } +\sqrt [ 3 ]{ x\left( x+h \right)  } +\sqrt [ 3 ]{ { x }^{ 2 } }  \right)  }  } =\frac { 1 }{ 3\sqrt [ 3 ]{ { x }^{ 2 } }  } $$
</s>