I've got a few criteria and I need to find a simple (as simple as possible) function which fits those criteria:
$\lim_{x \to -\infty }{f(x)} = C_1$ $f(0) = C_2$ f'(0) = C_3
Where $C_1,C_2,C_3$ are constants... ($C_1$ will be $1$, and the other constants are determined from another function so the function is continuous).
The function only has to be defined to $x = 0$. (As for the positive $x$ another function would be used).
I'm thinking along the line of:
$f(x) = \frac{ 1 }{ax + b} + c$
However I'm more or less struck on "using" the limit constraint to solve this problem.
$f(0) = \frac{1}{b} + c = C_2$ f'(0) = - \frac{a}{b^2} = C_3
Thanks in advance, paul23