The phrase, “…out on a limb” is actually only about 140 years old and it is an idiom that came from America. It is used in ways such as “I’m going out on a limb for you” or “You are going way out on a limb”. The meaning is taking a huge risk, usually on purpose. The first known use of the phrase was in the late 1880’s.
For example, if a person loans a large sum of money to someone who has no income but who promises to pay them back, the lender could be said to be going out on a limb.
The phrase is in reference to tree limbs, such as climbing a tree and getting out on one of the limbs. A person who did this would obviously be in a rather precarious situation. Should the limb break, they would fall out of the tree.
It is also easy to see that the higher up in the tree or the farther out on a branch a person gets, the greater the risk of something bad happening. This means that “You are going way out on a limb” is figuratively saying that you are taking a really big risk.
As idioms go, this one is pretty straightforward. Yet there are still people who try to draw a connection between arms and legs (body limbs) or think that the reference is to animals getting out on limbs. However, such connections are purely speculations and they aren’t supported by the first use of the phrase.
At any rate, you now know where ‘out on a limb’ comes from and what it means. Our language is rather colorful, isn’t it?