If $f:R\to R$ is uniformly continuous so $ \lim_{x \to \infty} f(\sqrt {x^2+5})-f(x) = 0$?

Question

Prove that if $f:R\to R$ is uniformly continuous so

$$ \lim_{x \to \infty} f(\sqrt {x^2+5})-f(x) = 0$$

any ideas?

thanks

Answer 1

$$\sqrt{x^2+5}-x = \frac{5}{x+\sqrt{x^2+5}} = O\left(\frac{1}{x}\right).\tag{1}$$ Uniform continuity gives that for any $\epsilon >0$ there is a $\delta>0$ such that $|x-y|\leq \delta$ implies $|f(x)-f(y)|\leq\epsilon$. As $x$ tends to infinity, the LHS of $(1)$ becomes arbitrarily small in absolute value, hence the wanted limit is just zero.