# The Convergence Behavior of a One-Parameter Family

The Convergence Behavior of a One-Parameter Family

Let (x)=, for . For any , (x)=e. Moreover:

f

α

x+α

1+

1

x

x>0

α∈

lim

x∞

f

α

1. If , then is strictly increasing.

α≤0

f

α

2. If , then is strictly decreasing.

α≥

1

2

f

α

3. If , then has exactly one local minimum and it is the absolute minimum.

0<α<

1

2

f

α