Consider function $f:\mathbb{R}^n \setminus \{0\} \times \mathbb{R}^n \rightarrow \mathbb{R}_{> 0}$ that is:
continuous in the first argument;
locally bounded in the second argument.
Consider a sequence $\{x_i\}_{i=1}^{\infty}$ such that $x_i \in \mathbb{R}^n$, $x_i \rightarrow x$.
Prove that
$$ \limsup_{i \rightarrow \infty} \ f(x,x_i) - f(x_i,x_i) \ = \ 0. $$