limit to the power limit

Let the above limit be $Y$

$Y = \displaystyle \lim _{x \rightarrow \infty} \begin{pmatrix} (x +1)^{\frac{1 + \frac 2x}{1 + \frac 1x}} - (x) ^{\frac {1 + \frac 1x}{1}} \end{pmatrix}$

$Y = \displaystyle \lim _{x \rightarrow \infty} \begin{pmatrix} ( x +1)^{\frac{ 1 + \frac 2\infty}{1 + \frac 1\infty}} - (x)^{\frac {1 + \frac 1\infty}{1}} \end{pmatrix}$

$Y = ((x +1)^{\frac 11} - (x) ^{\frac 11}) = (x +1 - x) = \boxed {1}$