<s>[INST] Limit with n roots I have been trying to practice computing limits and this one came across:
$$\lim_{x\to a} \frac{\sqrt[n]{x}-\sqrt[n]{a}}{x-a}$$
I tried L'Hopital and I got this:
$$\lim_{x\to a}{\frac{x^{\frac{1-2n}{n}}\left(1-n\right)}{n}}$$ 
But I should get as the solution of the limit $\frac{\sqrt[n]{a}}{an}$
Any help?
 [/INST] $$\lim _{ x\to a } \frac { \sqrt [ n ]{ x } -\sqrt [ n ]{ a }  }{ x-a } =\lim _{ x\to a } \frac { \left( \sqrt [ n ]{ x } -\sqrt [ n ]{ a }  \right) \left( \sqrt [ n ]{ { x }^{ n-1 } } +\sqrt [ n ]{ { x }^{ n-2 }a } +\sqrt [ n ]{ { x }^{ n-3 }{ a }^{ 2 } } +...+\sqrt [ n ]{ { a }^{ n-1 } }  \right)  }{ \left( x-a \right) \left( \sqrt [ n ]{ { x }^{ n-1 } } +\sqrt [ n ]{ { x }^{ n-2 }a } +\sqrt [ n ]{ { x }^{ n-3 }{ a }^{ 2 } } +...+\sqrt [ n ]{ { a }^{ n-1 } }  \right)  } =\\ =\lim _{ x\to a } \frac { 1 }{ \left( \sqrt [ n ]{ { x }^{ n-1 } } +\sqrt [ n ]{ { x }^{ n-2 }a } +\sqrt [ n ]{ { x }^{ n-3 }{ a }^{ 2 } } +...+\sqrt [ n ]{ { a }^{ n-1 } }  \right)  } =\frac { 1 }{ n\sqrt [ n ]{ { a }^{ n-1 } }  } =\frac { \sqrt [ n ]{ a }  }{ na } \\ $$
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