<s>[INST] Limit of $\mathrm{e}^{\sqrt{x+1}} - \mathrm{e}^{\sqrt{x}}$ How can I calculate the below limit? 
$$
\lim\limits_{x\to \infty} \left( \mathrm{e}^{\sqrt{x+1}} - \mathrm{e}^{\sqrt{x}} \right)
$$
In fact I know should use the L’Hospital’s Rule, but I do not how to use it.
 [/INST] Using the fact that $\lim _{ x\rightarrow 0 }{ \frac { { e }^{ x }-1 }{ x }  } =1\\ \\ $ we can write 
$$\lim _{ x\rightarrow \infty  }{ \left( e^{ \sqrt { x+1 }  }-e^{ \sqrt { x }  } \right)  } =\lim _{ x\rightarrow \infty  }{ { e }^{ \sqrt { x }  }\left( e^{ \sqrt { x+1 } -\sqrt { x }  }-1 \right)  } =\lim _{ x\rightarrow \infty  }{ { e }^{ \sqrt { x }  }\left( \frac { e^{ \frac { 1 }{ \sqrt { x+1 } +\sqrt { x }  }  }-1 }{ \frac { 1 }{ \sqrt { x+1 } +\sqrt { x }  }  }  \right)  } \cdot \frac { 1 }{ \sqrt { x+1 } +\sqrt { x }  } =\\ =\lim _{ x\rightarrow \infty  }{ \frac { { e }^{ \sqrt { x }  } }{ \sqrt { x+1 } +\sqrt { x }  }  } =+\infty $$
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