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Hess' Law

If 0<x<y th...

Question

If $0 < x < y$ then ${lim}_{n \to + \infty} {(y^{n} + x^{n})}^{1 / n}$

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Solution

$\begin{matrix} {(x^{n} + y^{n})}^{\frac{1}{n}} = y {(1 + \in^{n})}^{\frac{1}{h}} \Rightarrow t a k e f i x e d i \cdot i < δ \leq E [E = \frac{x}{y}] \Rightarrow E^{_{n}} \leq δ, f o r a l l n \geq N \Rightarrow 1 < {(1 + E^{n})}^{\frac{1}{n}} \leq {(1 + δ)}^{\frac{1}{n}} f o r n \geq N \Rightarrow {lim}_{x \to \infty} {(1 + δ)}^{\frac{1}{n}} = 1, {lim}_{x \to \infty} \frac{1}{n} ln (1 + δ) = 0 \Rightarrow {lim}_{x \to \infty} {(x^{n} + y^{h})}^{\frac{1}{n}} = y {lim}_{x \to \infty} (1 + δ) = y \end{matrix}$

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