You can choose 0 as the value of X in Rampaging War Mammoth's cycling cost. In that case, its triggered ability won't have any targets and won't destroy any artifacts.

You can cycle a card even if it has a triggered ability from cycling that won't have a legal target. This is because the cycling ability and the triggered ability are separate. This also means that if either ability is countered, the other ability will still resolve.

Triggered abilities from cycling a card and the cycling ability itself aren't spells. Effects that interact with spells (such as Saruman's Trickery) won't affect them.

Some cards with cycling have an ability that triggers when you cycle them, and some cards have an ability that triggers whenever you cycle any card. These triggered abilities resolve before you draw from the cycling ability.