## 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:

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 $$n^5$$?