Consider the mapping $x \mapsto f(x)$ with $f(x)=1+\|x\|_p$ for $x \in \mathbb{R}_{+}^n \smallsetminus\{0\},$ as $x_i>0$ for $i=1, \ldots, n,$ and $1 \leq p<\infty.$
Compute $\kappa_{\text{abs},\|\cdot\|_\infty}(x ; f)$ and $\kappa_{\text{rel},\|\cdot\|_\infty}(x ; f).$
Could someone please help me doing that? I don't know how to solve this problem.
$$ \kappa_{\mathrm{abs}}=\limsup _{\vec{x} \rightarrow x} \frac{\|f(\tilde{x})-f(x)\|}{\|\tilde{x}-x\|} $$ $$ \begin{aligned} \kappa_{\mathrm{rel}} &=\limsup _{\tilde{x} \rightarrow \alpha} \frac{\|f(\tilde{x})-f(x)\|}{\|\tilde{x}-x\|} \\ \kappa_{\mathrm{rel}} &=\frac{\|D f(x)\|\|x\|}{\|f(x)\|} \end{aligned} $$