Why is for close to ?
I wanted to see what the behaviour of the steep part of the curve was like, i.e. the behaviour of . So by thinking about a shift of the graph of to the left, I put some small (positive and negative) values of into my calculator for the function
for close to ?
or, in more colloquial terms,
The steep part of is just like the steep part of
But why is this the case? I couldn't deduce it easily using the Maclaurin expansion of . Is there a more intuitive explanation? I couldn't think of any explanations analogous to those explaining small angle approximations.