<s>[INST] How to solve $\lim\limits_{x \to -\infty} \left(x\left(\sqrt{x^2-x}-\sqrt{x^2-1}\right)\right)$? I have a problem with this limit, i have no idea how to compute it.
Can you explain the method and the steps used?
$$\lim\limits_{x \to -\infty} \left(x\left(\sqrt{x^2-x}-\sqrt{x^2-1}\right)\right)$$
 [/INST] $$\lim _{ x\to -\infty  } \left( \frac { x\left( \sqrt { x^{ 2 }-x } -\sqrt { x^{ 2 }-1 }  \right) \sqrt { x^{ 2 }-x } +\sqrt { x^{ 2 }-1 }  }{ \sqrt { x^{ 2 }-x } +\sqrt { x^{ 2 }-1 }  }  \right) =\\ =\lim _{ x\to -\infty  } \left( \frac { x\left( x^{ 2 }-x-x^{ 2 }+1 \right)  }{ \sqrt { x^{ 2 }-x } +\sqrt { x^{ 2 }-1 }  }  \right) =\lim _{ x\to -\infty  } \left( \frac { x\left( 1-x \right)  }{ \sqrt { x^{ 2 }-x } +\sqrt { x^{ 2 }-1 }  }  \right) =\\ \\ =\lim _{ x\to -\infty  } \frac { x\left( 1-x \right)  }{ \left| x \right| \left( \sqrt { 1-\frac { 1 }{ x }  } +\sqrt { 1-\frac { 1 }{ { x }^{ 2 } }  }  \right)  } =\\ \lim _{ x\to -\infty  } \frac { x-1 }{ \left( \sqrt { 1-\frac { 1 }{ x }  } +\sqrt { 1-\frac { 1 }{ { x }^{ 2 } }  }  \right)  } =-\infty  $$
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