Is the language $L=\{a^nb^m \mid n>2^m\}$ context-free?
According to Parikh's theorem, if your language were context-free, then the set S = $\{(n,m) : n &gt; 2^m\}$ would be semi-linear. Every linear subset of $S$ has a constant second coordinate, as we show below. Therefore you need an infinite number of linear sets to cover $S$, showing that it is not semi-linear.

It remains to show that every linear subset of $S$ has constant second coordinate. Indeed, consider any linear subset of $S$. By definition, such a subset is of the form $u_0 + \mathbb{N} u_1 + \cdots + \mathbb{N} u_r$. By assumption there exists some $i &gt; 0$ such that $u_i = (a,b)$, where $b \neq 0$. Let $u_0 = (n',m')$. It is not hard to check that for some $t$, $n' + at \leq 2^{m' + tb}$, since the left-hand side grows linearly, whereas the right-hand side grows exponentially. This contradicts the fact that the subset is a subset of $S$.