Limit by Squeeze Theorem - Part 1

$\begin{aligned} \left( n^2+{\color{#3D99F6} 0} \right)^\frac18 - (n+1)^\frac14 &< \left( n^2 +1 \right)^\frac18 - (n+1)^\frac14 < \left( n^2 {\color{#3D99F6} +2n}+1 \right)^\frac18 - (n+1)^\frac14 \\ n^\frac14 - (n+1)^\frac14 &< \left( n^2 +1 \right)^\frac18 - (n+1)^\frac14 < (n+1)^\frac14 - (n+1)^\frac14 \end{aligned}$

Since the right-hand side is zero for all $n$ , and the left-hand side tends to zero as $n \to \infty$ , by the squeeze theorem

$\lim_{n \to \infty} \left( n^2 +1 \right)^\frac18 - (n+1)^\frac14 = \boxed0$