Is $(n^5 + n^7)\in \Omega(n^7)$? Shouldn’t it be in $\Omega(n^5)$?

Is $ (n^5 + n^7)\in \Omega(n^7)$ ? Shouldn’t it be in $ \Omega(n^5)$ ?

I understand Omega to be a “lower bound” on a function. Shouldn’t the largest lower bound on the function $ n^5 + n^7$ be $ n^5$ ? (Just as the smallest upper bound is $ n^7$ )

The reason I say that the function is in the Omega class of functions larger than $ n^5$ is because of the limit definitions of complexity: enter image description here

Using the limit definition of $ O$ , we correctly identify that $ n^5+n^7 \in O(n^7)$ and $ n^5+n^7 \notin O(n^6)$

However, using the limit definition of $ \Omega$ tells us that the function is in $ \Omega(n^7)$ ! But isn’t the largest lower bound for this function $ n^5$ ?