Uniform continuity on $(0, \infty)$ and limits of the function at $\lim$ at $0$ and $\infty$

Question

Are the following statements True or False?

1. If $ f : (0, \infty) \rightarrow \mathbb{R}$ is uniformly continuous then $\lim_{x \rightarrow 0} f(x)$ exists.

2. If $ f : (0, \infty) \rightarrow \mathbb{R}$ is uniformly continuous then $\lim_{x \rightarrow \infty} f(x)$ exists.

For part 1, this is what I tried : consider $\{x_n\}$, $\{y_n\}$ be two Cauchy sequences converging to 0. Then by uniform continuity $\{f(x_n)\}$ and $\{f(y_n)\}$ are Cauchy and convergent to the same point $l = \lim_{n \rightarrow \infty} f(x_n)$. To show $l = f(\lim_{n \rightarrow \infty} x_n) = f(0)$, which is satisfied as f is continuous on $(0, \infty)$ hence the limits can be exchanged. Hence $\lim_{x \rightarrow 0} f(x)$ exists.

For the second part, by uniform continuity on $[b, \infty)$, $b$ sufficiently large, given $\epsilon > 0$, there exists $\delta > 0$ such that whenever $|x-y| < \delta$, $|f(x)- f(y)| < \epsilon$. Consider a sequence $x_n \rightarrow \infty$. I am stuck here.

Could you please check my ideas and help me finish the problems? Thank you.

Answer 1

The first assertion is indeed true. The second one is false. Take the sine function, for instance.