Evaluate using Taylor polynomials: lim as x to infinity

Evaluate using Taylor polynomials: $\displaystyle\lim_{x\to\infty}\Bigl(\sqrt[2n]{x^{2n}+x^{2n-1}}-\sqrt[2n]{x^{2n}-x^{2n-1}}\Bigr)$.

Step 1: Factor $x$ out of each radical

\[ \sqrt[2n]{x^{2n}\pm x^{2n-1}} = x\left(1\pm\frac1x\right)^{1/(2n)}. \tag{1} \]

The expression becomes $x\left[\left(1+\tfrac1x\right)^{1/(2n)}-\left(1-\tfrac1x\right)^{1/(2n)}\right]$.

Step 2: Expand with $t=1/x\to0^+$

\[ \left(1\pm\frac1x\right)^{1/(2n)} = 1\pm\frac1{2nx}+o\!\left(\frac1x\right). \tag{2} \]

Step 3: Subtract and multiply by $x$

\[ \left(1+\frac1x\right)^{1/(2n)}-\left(1-\frac1x\right)^{1/(2n)} = \frac1{nx}+o\!\left(\frac1x\right) \;\Longrightarrow\; x\left[\frac1{nx}+o\!\left(\frac1x\right)\right] = \frac1n+o(1) \xrightarrow[x\to\infty]{} \frac1n. \tag{3} \]

Step 4: Verification

\[n=2,\ x=100000\text{: numeric value}\approx0.50000000001,\ \text{matching }1/n=0.5\ \checkmark\]