How to solve this limit problem? $\lim_{n \to \infty} \left ( \sqrt{n + \sqrt{n}} - \sqrt{n - \sqrt{n}} \right )$ I am stuck with the following question from my homework:
$\lim_{n \to \infty} \left ( \sqrt{n + \sqrt{n}} - \sqrt{n - \sqrt{n}} \right )$
Using wolfram alpha gives me 1 for the solution. However, I would like to know how you come up with this result. I hope someone can explain this to me.

$$\lim _{ n\to \infty  } \left( \sqrt { n+\sqrt { n }  } -\sqrt { n-\sqrt { n }  }  \right) =\lim _{ n\to \infty  } \frac { \left( \sqrt { n+\sqrt { n }  } -\sqrt { n-\sqrt { n }  }  \right) \left( \sqrt { n+\sqrt { n }  } +\sqrt { n-\sqrt { n }  }  \right)  }{ \left( \sqrt { n+\sqrt { n }  } +\sqrt { n-\sqrt { n }  }  \right)  }\\\\ =\lim _{ n\to \infty  } \frac { 2\sqrt { n }  }{ \left( \sqrt { n+\sqrt { n }  } +\sqrt { n-\sqrt { n }  }  \right)  } =\lim _{ n\to \infty  } \frac { 2\sqrt { n }  }{ \sqrt { n } \left( \sqrt { 1+\frac { 1 }{ \sqrt { n }  }  } +\sqrt { 1-\frac { 1 }{ \sqrt { n }  }  }  \right)  } =\\=\lim _{ n\to \infty  } \frac { 2 }{ \left( \sqrt { 1+\frac { 1 }{ \sqrt { n }  }  } +\sqrt { 1-\frac { 1 }{ \sqrt { n }  }  }  \right)  } =1$$
