Convergence of a sub-sequence

Question

I am having a doubt about the convergence of a sub-sequence as follows.
Knowing that sequences $x(k),y(k) \in \mathbb{R}^n$ converge to $x^*, y^*$ respectively as $k$ goes to $0$ ($k \in \mathbb{R}^+$). We also have $<x(k), y(k)> = k$. So, $<x^*, y^*> = 0$.
Now I divide both sides of the inner product by $k$ and consider $<x(k), \frac{y(k)}{k}> = 1$.
My question is when $k$ goes to $0$, whether I can prove the limit of $\frac{y(k)}{k}$ to a certain point $\hat{y}$?