https://www.reddit.com/r/askmath/comments/d79rp0/can_someone_help_me_with_this_question/

"When in doubt L'Hopital"

$$ (\Pi_{n=0}^{N} (x+n))^{1/N} - x $$ distribute the product $$ (\Sigma_{n=0}^{N} a_n x^n)^{1/N} - x $$

multiply and divide by x $$ \frac {(\Sigma a_n x^{n-N})^{1/N} - 1} {1/x} $$

l'hopital $$ \frac{(1/N)(\Sigma a_n x^{n-N})^{1/N-1} (\Sigma (n-N)a_n x^{n-N-1})} {-1/x^2} $$

multiply and divide by x^2 $$ (-1/N)(\Sigma a_n x^{n-N})^{1/N-1} (\Sigma (n-N)a_n x^{n-N+1}) $$

distribute the limit according to limit laws $$ (\lim -1/N)(\lim \Sigma a_n x^{n-N})^{1/N-1}(\lim \Sigma (n-N) a_n x^{n-N+1}) $$

only the strongest survive (x^0 terms)

$$ (-1/N)(a_N)^{1/N-1}(((N-1)-N) a_{N-1}) $$

$$ \frac { (a_N)^{1/N-1} a_{N-1} } {N}$$

You can work out the terms but these particular ones also have formulas. If it were other ones it would be harder and require pascal's triangle (binomial theorem).

$$ \frac {1 \cdot N(N+1)/2 } {N}$$

$$ \frac { 1 \cdot 15 } {5} = 3 $$