My objection was to saying $\lim \left[7^{n}\left\{\left(\frac{-3}{7}\right)^{
n}+\left(\frac{-5}{7}\right)^{n}+1\right\}\right]^{\frac{1}{n}} = \lim \left[7^{n}\left\{\left(0 + 0 +1\right\}\right]^{\frac{1}{n}}$ on the grounds that the inner limits go to zero. If that sort of logic applied, one might claim that $\lim (1 + \frac1n)^n = \lim (1 + 0)^n = 1$ rather than $e$.
n}+\left(\frac{-5}{7}\right)^{n}+1\right\}\right]^{\frac{1}{n}} = \lim \left[7^{n}\left\{\left(0 + 0 +1\right\}\right]^{\frac{1}{n}}$ on the grounds that the inner limits go to zero. If that sort of logic applied, one might claim that $\lim (1 + \frac1n)^n = \lim (1 + 0)^n = 1$ rather than $e$.
