Q:

Question regarding the value of $y$ in this inequality $|\lim\limits_{x\rightarrow\infty}\frac{x^{1-y}}{x^{y}}|<\infty$

Find all possible values of $y$ such that $$|\lim\limits_{x\rightarrow\infty}\frac{x^{1-y}}{x^{y}}|<\infty$$

A:

We can rewrite it as $|\lim_{x\to \infty} x^{1-2y}|$ and this limit is clearly less than infinity only when $1-2y≤0$. If $y>1/2$ the limit is zero and if $y=1/2$ the limit is $1$.

