The discussion focuses on determining the slope of functions at the edge case of $x=1$. Two functions are analyzed: $f(x)=x^2-1$ and $f(x)=2x+1$. For $f(x)=x^2-1$, the slope at $x=1$ is calculated, and it is confirmed that the graph includes the point $(1,0)$. Similarly, for $f(x)=2x+1$, the slope at $x=1$ is also evaluated, with a check on whether the graph includes the point $(1,0)$. Understanding these slopes at the edge case is crucial for analyzing function behavior.