# Looking for a multivariable function with an assymptote and a gradient bounded away from zero.

I am Looking for a multivariable function with an assymptote and a gradient bounded away from zero.

For example: let $f:\mathbb{R}^k\rightarrow\mathbb{R}$ be a function such that

1. $lim f(x_1,\cdots x_k)\rightarrow M$ where $M$ is a finite constant
2. $\forall x\in \mathbb{R}^k$, $\nabla f>\delta(x)$ where $\delta(x)>0$.
• What do you mean by an asymptote in a multivariable function? Anyway, if you want $\lim_{|\mathbf{x}| \to \infty} f( \mathbf{x} ) = M < \infty$, then necessarily the gradient must approach to $0$, so that such a function cannot exist. – Crostul Oct 30 '16 at 12:53
• Is $f(\mathbf{x})=\sum_{i=1}^kx_i$ not such a function? It has a constant gradient, not equal to $0$, and is its own asymptote. – ahorn Oct 30 '16 at 13:01
• I am looking for a function that is never " too close" to the assymptote, while by defn. this distance (i.e., the gradient) will approach zero, I want it to "reach" 0 only in the limit... – MoRkO Nov 1 '16 at 9:33