WOLFRAM|DEMONSTRATIONS PROJECT

The Convergence Behavior of a One-Parameter Family

α

0.1

x+0.1

1

x

+1

Let

f

α

(x)=

x+α

1+

1

x

, for

x>0

. For any

α∈

,

lim

x∞

f

α

(x)=e

. Moreover:

1. If

α≤0

, then

f

α

is strictly increasing.

2. If

α≥

1

2

, then

f

α

is strictly decreasing.

3. If

0<α<

1

2

, then

f

α

has exactly one local minimum and it is the absolute minimum.