Elizabeth and Zachariah had obviously given up hope of having a baby. She was probably about 60yrs and so what was the point of continuing to believe.
But they both kept on loving God and living blamelessly for Him. Zachariah diligently kept at his priestly duties. And one day God showed up while he was serving. God's angel came in Lk 1:13 and said:
"Do not be afraid, Zachariah, for your prayer is heard and your wife Elizabeth will bear you a son..."
And 9 months later, they did have a baby.
The key lesson to learn here is that God is not limited by the passing of time. As we diligently do his work, he will surely reward and we can keep on believing that He is able and willing to do exceedingly more than we can ever ask or think.