Limit of difference of roots

$\large \displaystyle L = \lim_{n \rightarrow \infty} n \left [\sqrt[7]{n^7+7n^6} - \sqrt{n^2+2n} \right ]$

$\large \displaystyle L = \lim_{n \rightarrow \infty} n^2 \left [ \left(1+\frac7n \right)^{\frac17} - \left(1+\frac2n \right)^{\frac12} \right ]$

Using the formula for binomial expansion of $(1+x)^p$ :

$\large \displaystyle L = \lim_{n \rightarrow \infty} n^2 \left [ \left(1+\frac17 \cdot \frac7n - \frac{3}{49}\cdot \left(\frac 7n \right )^2 + \mathbb{O}(n^{-3}) \right ) - \left(1+ \frac12 \cdot \frac 2n - \frac18 \cdot \left ( \frac 2n \right) ^2 + \mathbb{O}(n^{-3}) \right) \right ]$

$\large \displaystyle L = \lim_{n \rightarrow \infty} n^2 \left [ \frac{1}{2n^2} - \frac{3}{n^2} + \mathbb{O}(n^{-3}) \right]$

$\large \displaystyle L = \lim_{n \rightarrow \infty} \left [ \frac12- 3 + \mathbb{O}(n^{-1}) \right]$

$\color{#3D99F6} \boxed{\large \displaystyle L = - \frac52 = -2.5}$