Let $f:\mathbb{Q}\to \mathbb{R}$, $f(p/q)=a^{p/q}$ such that $a>1$. Prove that $\lim_{x\to 0}f(x)=1$

Question

I need to solve that problem. I tried proving instead $\lim_{x\to \infty}a^{1/x}=1$. But the complete proble say: Let $f:\mathbb{Q}\to \mathbb{R}$, with $f(p/q)=a^{p/q}$ such that a>1. Prove that $\lim_{x\to 0}f(x)=1$ and conclude:

1. For all $b\in\mathbb{R}$, exists $\lim_{x\to b} f(x)$.
2. If $b\in \mathbb{Q}$ then $\lim_{x\to b} f(x)=f(b)$.
3. $a^x \cdot a^y =a^{x+y}$
4. If $x<y$ then $a^x < a^y$.

I use the fourth point but the problem tells me that i need to conclude it from the first statement. I really don't know how to proceed, may something like $\varepsilon -\delta$. I'll apreciate your help.