A question about the behavior of the algebraic curve $y=x^n$

Question

The algebraic curve $y=x^n$ shows steep rise in value as $x$ become closer to 1.

How to find the point at which the steep rise starts?

Answer 1

What, exactly, is a steep rise?

If you define a steep rise as greater than a slope of $m$,then this is not too hard to find.

For $y=x^n$,the derivative, or tangent slope, at any point is $nx^{n-1}$.

If $nx^{n-1}\geq m$, then $\displaystyle x \geq \sqrt[n-1]{\frac{m}{n}}$

Answer 2

Such a point is not defined, it is a subjective concept.