Limit of function with natural logarithm I need help solving this problem. I tried L'hospital and rearranging but nothing worked.
$$
\lim_\limits{x→∞} x^2\left(\ln\left(1 + \frac{1}{x}\right)- \frac{1}{x+1}\right)
$$

Using L'Hospital rule we get $$\\ \lim _{ x\rightarrow \infty  }{ \frac { \ln { \left( 1+\frac { 1 }{ x }  \right) -\frac { 1 }{ x+1 }  }  }{ \frac { 1 }{ { x }^{ 2 } }  }  } =\lim _{ x\rightarrow \infty  }{ \frac { \frac { x }{ x+1 } \cdot \left( -\frac { 1 }{ { x }^{ 2 } }  \right) +\frac { 1 }{ { \left( x+1 \right)  }^{ 2 } }  }{ \frac { -2 }{ { x }^{ 3 } }  }  } =\\ =\lim _{ x\rightarrow \infty  }{ \frac { -x-1+x }{ { x\left( x+1 \right)  }^{ 2 } } \cdot \frac { { x }^{ 3 } }{ -2 }  } =\lim _{ x\rightarrow \infty  }{ \frac { 1 }{ { \left( x+1 \right)  }^{ 2 } } \cdot \frac { { x }^{ 2 } }{ 2 }  } =\frac { 1 }{ 2 } $$
